Theorem ufli 21718

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

Assertion

Ref	Expression
ufli	|- ( ( X e. UFL /\ F e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) F C_ 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 21717	. . 3 |- ( X e. UFL -> ( X e. UFL <-> A. g e. ( Fil ` X ) E. f e. ( UFil ` X ) g C_ f ) )
2	1	ibi 256	. 2 |- ( X e. UFL -> A. g e. ( Fil ` X ) E. f e. ( UFil ` X ) g C_ f )
3		sseq1 3626	. . . 4 |- ( g = F -> ( g C_ f <-> F C_ f ) )
4	3	rexbidv 3052	. . 3 |- ( g = F -> ( E. f e. ( UFil ` X ) g C_ f <-> E. f e. ( UFil ` X ) F C_ f ) )
5	4	rspccva 3308	. 2 |- ( ( A. g e. ( Fil ` X ) E. f e. ( UFil ` X ) g C_ f /\ F e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) F C_ f )
6	2, 5	sylan 488	1 |- ( ( X e. UFL /\ F e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) F C_ f )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ` cfv 5888   Filcfil 21649   UFilcufil 21703  UFLcufl 21704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ufl 21706

This theorem is referenced by:  ssufl  21722  ufldom  21766  ufilcmp  21836