Theorem ufli 23829

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 23828	. . 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 3955	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑓 ↔ 𝐹 ⊆ 𝑓))
4	3	rexbidv 3156	. . 3 ⊢ (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓))
5	4	rspccva 3571	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ‘cfv 6481  Filcfil 23760  UFilcufil 23814  UFLcufl 23815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ufl 23817

This theorem is referenced by:  ssufl  23833  ufldom  23877  ufilcmp  23947