Theorem qtopres 21501

Description: The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that F be a function with domain X. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

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

Assertion

Ref	Expression
qtopres	|- ( F e. V -> ( J qTop F ) = ( J qTop ( F |` X ) ) )

Proof of Theorem qtopres

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

Step	Hyp	Ref	Expression
1		resima 5431	. . . . . . 7 |- ( ( F |` X ) " X ) = ( F " X )
2	1	pweqi 4162	. . . . . 6 |- ~P ( ( F |` X ) " X ) = ~P ( F " X )
3		rabeq 3192	. . . . . 6 |- ( ~P ( ( F |` X ) " X ) = ~P ( F " X ) -> { s e. ~P ( ( F |` X ) " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J } = { s e. ~P ( F " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J } )
4	2, 3	ax-mp 5	. . . . 5 |- { s e. ~P ( ( F |` X ) " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J } = { s e. ~P ( F " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J }
5		residm 5430	. . . . . . . . . . 11 |- ( ( F |` X ) |` X ) = ( F |` X )
6	5	cnveqi 5297	. . . . . . . . . 10 |- `' ( ( F |` X ) |` X ) = `' ( F |` X )
7	6	imaeq1i 5463	. . . . . . . . 9 |- ( `' ( ( F |` X ) |` X ) " s ) = ( `' ( F |` X ) " s )
8		cnvresima 5623	. . . . . . . . 9 |- ( `' ( ( F |` X ) |` X ) " s ) = ( ( `' ( F |` X ) " s ) i^i X )
9		cnvresima 5623	. . . . . . . . 9 |- ( `' ( F |` X ) " s ) = ( ( `' F " s ) i^i X )
10	7, 8, 9	3eqtr3i 2652	. . . . . . . 8 |- ( ( `' ( F |` X ) " s ) i^i X ) = ( ( `' F " s ) i^i X )
11	10	eleq1i 2692	. . . . . . 7 |- ( ( ( `' ( F |` X ) " s ) i^i X ) e. J <-> ( ( `' F " s ) i^i X ) e. J )
12	11	a1i 11	. . . . . 6 |- ( s e. ~P ( F " X ) -> ( ( ( `' ( F |` X ) " s ) i^i X ) e. J <-> ( ( `' F " s ) i^i X ) e. J ) )
13	12	rabbiia 3185	. . . . 5 |- { s e. ~P ( F " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J } = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J }
14	4, 13	eqtr2i 2645	. . . 4 |- { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } = { s e. ~P ( ( F |` X ) " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J }
15		qtopval.1	. . . . 5 |- X = U. J
16	15	qtopval 21498	. . . 4 |- ( ( J e. _V /\ F e. V ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )
17		resexg 5442	. . . . 5 |- ( F e. V -> ( F |` X ) e. _V )
18	15	qtopval 21498	. . . . 5 |- ( ( J e. _V /\ ( F |` X ) e. _V ) -> ( J qTop ( F |` X ) ) = { s e. ~P ( ( F |` X ) " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J } )
19	17, 18	sylan2 491	. . . 4 |- ( ( J e. _V /\ F e. V ) -> ( J qTop ( F |` X ) ) = { s e. ~P ( ( F |` X ) " X ) | ( ( `' ( F |` X ) " s ) i^i X ) e. J } )
20	14, 16, 19	3eqtr4a 2682	. . 3 |- ( ( J e. _V /\ F e. V ) -> ( J qTop F ) = ( J qTop ( F |` X ) ) )
21	20	expcom 451	. 2 |- ( F e. V -> ( J e. _V -> ( J qTop F ) = ( J qTop ( F |` X ) ) ) )
22		df-qtop 16167	. . . . 5 |- qTop = ( j e. _V , f e. _V |-> { s e. ~P ( f " U. j ) | ( ( `' f " s ) i^i U. j ) e. j } )
23	22	reldmmpt2 6771	. . . 4 |- Rel dom qTop
24	23	ovprc1 6684	. . 3 |- ( -. J e. _V -> ( J qTop F ) = (/) )
25	23	ovprc1 6684	. . 3 |- ( -. J e. _V -> ( J qTop ( F |` X ) ) = (/) )
26	24, 25	eqtr4d 2659	. 2 |- ( -. J e. _V -> ( J qTop F ) = ( J qTop ( F |` X ) ) )
27	21, 26	pm2.61d1 171	1 |- ( F e. V -> ( J qTop F ) = ( J qTop ( F |` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   `'ccnv 5113   |` cres 5116   "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:  qtoptop2  21502