Theorem ufli 21528

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 21527	. . 3 ⊢ (𝑋 ∈ UFL → (𝑋 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓))
2	1	ibi 255	. 2 ⊢ (𝑋 ∈ UFL → ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
3		sseq1 3589	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑓 ↔ 𝐹 ⊆ 𝑓))
4	3	rexbidv 3034	. . 3 ⊢ (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓))
5	4	rspccva 3281	. 2 ⊢ ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓 ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
6	2, 5	sylan 487	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  Filcfil 21459  UFilcufil 21513  UFLcufl 21514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ufl 21516

This theorem is referenced by:  ssufl  21532  ufldom  21576  ufilcmp  21646