Theorems · Theorem · general topology
Filter.mem_prod_iff
∀ {α : Type u_1} {β : Type u_2} {s : Set (α × β)} {f : Filter α} {g : Filter β},
s ∈ f ×ˢ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s- Defined in
- Mathlib.Order.Filter.Prod
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Filterstatement and proof · cited by 8,121
- Set.preimageproof · cited by 4,946
- SProd.sprodstatement and proof · cited by 1,750
- Filter.comapproof · cited by 546
- Filter.preimage_mem_comapproof · cited by 23
- Filter.mem_inf_of_interproof · cited by 6
Cited by21
Results whose statement or proof uses this declaration.
- Filter.prod_map_map_eqproof · cited by 17
- Filter.eventually_prod_iffproof · cited by 12
- CauchyFilter.denseRange_pureCauchyproof · cited by 5
- isSeparatedMap_iff_isClosed_diagonalproof · cited by 4
- mem_nhds_prod_iffproof · cited by 4
- compact_open_separated_add_rightproof · cited by 3
- Filter.mem_prod_principalproof · cited by 3
- isLocallyInjective_iff_isOpen_diagonalproof · cited by 3
- compact_open_separated_mul_rightproof · cited by 2
- mem_nhdsWithin_prod_iffproof · cited by 2
- LaurentSeries.Cauchy.exists_lb_eventual_supportproof · cited by 2
- Filter.prod_mem_prod_iffproof · cited by 2