Theorem ufli 22524

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 22523	. . 3 ⊢ (𝑋 ∈ UFL → (𝑋 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓))
2	1	ibi 269	. 2 ⊢ (𝑋 ∈ UFL → ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
3		sseq1 3994	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑓 ↔ 𝐹 ⊆ 𝑓))
4	3	rexbidv 3299	. . 3 ⊢ (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓))
5	4	rspccva 3624	. 2 ⊢ ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
6	2, 5	sylan 582	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938  ‘cfv 6357  Filcfil 22455  UFilcufil 22509  UFLcufl 22510

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ufl 22512

This theorem is referenced by:  ssufl  22528  ufldom  22572  ufilcmp  22642