Theorem qtopval2 21499

Description: Value of the quotient topology function when F is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

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

Assertion

Ref	Expression
qtopval2	|- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( J qTop F ) = { s e. ~P Y | ( `' F " s ) e. J } )

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

Proof of Theorem qtopval2

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> J e. V )
2		fof 6115	. . . . 5 |- ( F : Z -onto-> Y -> F : Z --> Y )
3	2	3ad2ant2 1083	. . . 4 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> F : Z --> Y )
4		qtopval.1	. . . . . 6 |- X = U. J
5		uniexg 6955	. . . . . . 7 |- ( J e. V -> U. J e. _V )
6	5	3ad2ant1 1082	. . . . . 6 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> U. J e. _V )
7	4, 6	syl5eqel 2705	. . . . 5 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> X e. _V )
8		simp3 1063	. . . . 5 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> Z C_ X )
9	7, 8	ssexd 4805	. . . 4 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> Z e. _V )
10		fex 6490	. . . 4 |- ( ( F : Z --> Y /\ Z e. _V ) -> F e. _V )
11	3, 9, 10	syl2anc 693	. . 3 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> F e. _V )
12	4	qtopval 21498	. . 3 |- ( ( J e. V /\ F e. _V ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )
13	1, 11, 12	syl2anc 693	. 2 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( J qTop F ) = { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } )
14		imassrn 5477	. . . . . 6 |- ( F " X ) C_ ran F
15		forn 6118	. . . . . . 7 |- ( F : Z -onto-> Y -> ran F = Y )
16	15	3ad2ant2 1083	. . . . . 6 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ran F = Y )
17	14, 16	syl5sseq 3653	. . . . 5 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( F " X ) C_ Y )
18		foima 6120	. . . . . . 7 |- ( F : Z -onto-> Y -> ( F " Z ) = Y )
19	18	3ad2ant2 1083	. . . . . 6 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( F " Z ) = Y )
20		imass2 5501	. . . . . . 7 |- ( Z C_ X -> ( F " Z ) C_ ( F " X ) )
21	8, 20	syl 17	. . . . . 6 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( F " Z ) C_ ( F " X ) )
22	19, 21	eqsstr3d 3640	. . . . 5 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> Y C_ ( F " X ) )
23	17, 22	eqssd 3620	. . . 4 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( F " X ) = Y )
24	23	pweqd 4163	. . 3 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ~P ( F " X ) = ~P Y )
25		cnvimass 5485	. . . . . . 7 |- ( `' F " s ) C_ dom F
26		fdm 6051	. . . . . . . 8 |- ( F : Z --> Y -> dom F = Z )
27	3, 26	syl 17	. . . . . . 7 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> dom F = Z )
28	25, 27	syl5sseq 3653	. . . . . 6 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( `' F " s ) C_ Z )
29	28, 8	sstrd 3613	. . . . 5 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( `' F " s ) C_ X )
30		df-ss 3588	. . . . 5 |- ( ( `' F " s ) C_ X <-> ( ( `' F " s ) i^i X ) = ( `' F " s ) )
31	29, 30	sylib 208	. . . 4 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( ( `' F " s ) i^i X ) = ( `' F " s ) )
32	31	eleq1d 2686	. . 3 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( ( ( `' F " s ) i^i X ) e. J <-> ( `' F " s ) e. J ) )
33	24, 32	rabeqbidv 3195	. 2 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> { s e. ~P ( F " X ) | ( ( `' F " s ) i^i X ) e. J } = { s e. ~P Y | ( `' F " s ) e. J } )
34	13, 33	eqtrd 2656	1 |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( J qTop F ) = { s e. ~P Y | ( `' F " s ) e. J } )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   -onto->wfo 5886  (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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-qtop 16167

This theorem is referenced by:  elqtop  21500