Theorem ufli 23801

Description: Property of a set that satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
ufli	⊢ ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Distinct variable groups:   𝑓,𝐹   𝑓,𝑋

Proof of Theorem ufli

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isufl 23800	. . 3 ⊢ (𝑋 ∈ UFL → (𝑋 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓))
2	1	ibi 267	. 2 ⊢ (𝑋 ∈ UFL → ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
3		sseq1 3972	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑓 ↔ 𝐹 ⊆ 𝑓))
4	3	rexbidv 3157	. . 3 ⊢ (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓))
5	4	rspccva 3587	. 2 ⊢ ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
6	2, 5	sylan 580	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ‘cfv 6511  Filcfil 23732  UFilcufil 23786  UFLcufl 23787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ufl 23789

This theorem is referenced by:  ssufl  23805  ufldom  23849  ufilcmp  23919