Theorem qtopval 21498

Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

Ref	Expression
qtopval.1	|- X = U. J

Assertion

Ref	Expression
qtopval	|- ( ( J e. V /\ F e. W ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )

Distinct variable groups:   F, s   J, s   V, s   X, s

Allowed substitution hint:   W( s)

Proof of Theorem qtopval

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

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( J e. V -> J e. _V )
2		elex 3212	. 2 |- ( F e. W -> F e. _V )
3		imaexg 7103	. . . . 5 |- ( F e. _V -> ( F " X ) e. _V )
4		pwexg 4850	. . . . 5 |- ( ( F " X ) e. _V -> ~P ( F " X ) e. _V )
5		rabexg 4812	. . . . 5 |- ( ~P ( F " X ) e. _V -> { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } e. _V )
6	3, 4, 5	3syl 18	. . . 4 |- ( F e. _V -> { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } e. _V )
7	6	adantl 482	. . 3 |- ( ( J e. _V /\ F e. _V ) -> { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } e. _V )
8		simpr 477	. . . . . . 7 |- ( ( j = J /\ f = F ) -> f = F )
9		simpl 473	. . . . . . . . 9 |- ( ( j = J /\ f = F ) -> j = J )
10	9	unieqd 4446	. . . . . . . 8 |- ( ( j = J /\ f = F ) -> U. j = U. J )
11		qtopval.1	. . . . . . . 8 |- X = U. J
12	10, 11	syl6eqr 2674	. . . . . . 7 |- ( ( j = J /\ f = F ) -> U. j = X )
13	8, 12	imaeq12d 5467	. . . . . 6 |- ( ( j = J /\ f = F ) -> ( f " U. j ) = ( F " X ) )
14	13	pweqd 4163	. . . . 5 |- ( ( j = J /\ f = F ) -> ~P ( f " U. j ) = ~P ( F " X ) )
15	8	cnveqd 5298	. . . . . . . 8 |- ( ( j = J /\ f = F ) -> `' f = `' F )
16	15	imaeq1d 5465	. . . . . . 7 |- ( ( j = J /\ f = F ) -> ( `' f " s ) = ( `' F " s ) )
17	16, 12	ineq12d 3815	. . . . . 6 |- ( ( j = J /\ f = F ) -> ( ( `' f " s ) i^i U. j ) = ( ( `' F " s ) i^i X ) )
18	17, 9	eleq12d 2695	. . . . 5 |- ( ( j = J /\ f = F ) -> ( ( ( `' f " s ) i^i U. j ) e. j <-> ( ( `' F " s ) i^i X ) e. J ) )
19	14, 18	rabeqbidv 3195	. . . 4 |- ( ( j = J /\ f = F ) -> { s e. ~P ( f " U. j ) | ( ( `' f " s ) i^i U. j ) e. j } = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )
20		df-qtop 16167	. . . 4 |- qTop = ( j e. _V , f e. _V |-> { s e. ~P ( f " U. j ) | ( ( `' f " s ) i^i U. j ) e. j } )
21	19, 20	ovmpt2ga 6790	. . 3 |- ( ( J e. _V /\ F e. _V /\ { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } e. _V ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )
22	7, 21	mpd3an3 1425	. 2 |- ( ( J e. _V /\ F e. _V ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )
23	1, 2, 22	syl2an 494	1 |- ( ( J e. V /\ F e. W ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   `'ccnv 5113   "cima 5117  (class class class)co 6650   qTop cqtop 16163

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  ax-un 6949

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-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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-qtop 16167

This theorem is referenced by:  qtopval2  21499  qtopres  21501  imastopn  21523