Theorem ufli 23879

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 23878	. . 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 3947	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑓 ↔ 𝐹 ⊆ 𝑓))
4	3	rexbidv 3161	. . 3 ⊢ (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓))
5	4	rspccva 3563	. 2 ⊢ ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
6	2, 5	sylan 581	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889  ‘cfv 6498  Filcfil 23810  UFilcufil 23864  UFLcufl 23865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ufl 23867

This theorem is referenced by:  ssufl  23883  ufldom  23927  ufilcmp  23997