Theorem fgval 21674

Description: The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fgval	|- ( F e. ( fBas ` X ) -> ( X filGen F ) = { x e. ~P X | ( F i^i ~P x ) =/= (/) } )

Distinct variable groups:   x, F   x, X

Proof of Theorem fgval

Dummy variables v f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fg 19744	. . 3 |- filGen = ( v e. _V , f e. ( fBas ` v ) |-> { x e. ~P v | ( f i^i ~P x ) =/= (/) } )
2	1	a1i 11	. 2 |- ( F e. ( fBas ` X ) -> filGen = ( v e. _V , f e. ( fBas ` v ) |-> { x e. ~P v | ( f i^i ~P x ) =/= (/) } ) )
3		pweq 4161	. . . . 5 |- ( v = X -> ~P v = ~P X )
4	3	adantr 481	. . . 4 |- ( ( v = X /\ f = F ) -> ~P v = ~P X )
5		ineq1 3807	. . . . . 6 |- ( f = F -> ( f i^i ~P x ) = ( F i^i ~P x ) )
6	5	neeq1d 2853	. . . . 5 |- ( f = F -> ( ( f i^i ~P x ) =/= (/) <-> ( F i^i ~P x ) =/= (/) ) )
7	6	adantl 482	. . . 4 |- ( ( v = X /\ f = F ) -> ( ( f i^i ~P x ) =/= (/) <-> ( F i^i ~P x ) =/= (/) ) )
8	4, 7	rabeqbidv 3195	. . 3 |- ( ( v = X /\ f = F ) -> { x e. ~P v | ( f i^i ~P x ) =/= (/) } = { x e. ~P X | ( F i^i ~P x ) =/= (/) } )
9	8	adantl 482	. 2 |- ( ( F e. ( fBas ` X ) /\ ( v = X /\ f = F ) ) -> { x e. ~P v | ( f i^i ~P x ) =/= (/) } = { x e. ~P X | ( F i^i ~P x ) =/= (/) } )
10		fveq2 6191	. . 3 |- ( v = X -> ( fBas ` v ) = ( fBas ` X ) )
11	10	adantl 482	. 2 |- ( ( F e. ( fBas ` X ) /\ v = X ) -> ( fBas ` v ) = ( fBas ` X ) )
12		elfvex 6221	. 2 |- ( F e. ( fBas ` X ) -> X e. _V )
13		id 22	. 2 |- ( F e. ( fBas ` X ) -> F e. ( fBas ` X ) )
14		elfvdm 6220	. . 3 |- ( F e. ( fBas ` X ) -> X e. dom fBas )
15		pwexg 4850	. . 3 |- ( X e. dom fBas -> ~P X e. _V )
16		rabexg 4812	. . 3 |- ( ~P X e. _V -> { x e. ~P X | ( F i^i ~P x ) =/= (/) } e. _V )
17	14, 15, 16	3syl 18	. 2 |- ( F e. ( fBas ` X ) -> { x e. ~P X | ( F i^i ~P x ) =/= (/) } e. _V )
18	2, 9, 11, 12, 13, 17	ovmpt2dx 6787	1 |- ( F e. ( fBas ` X ) -> ( X filGen F ) = { x e. ~P X | ( F i^i ~P x ) =/= (/) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   fBascfbas 19734   filGencfg 19735

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fg 19744

This theorem is referenced by:  elfg  21675  restmetu  22375  neifg  32366