Theorem fparlem3 6901

Description: Lemma for fpar 6903. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fparlem3	|- ( F Fn A -> ( `' ( 1st |` ( _V X. _V ) ) o. ( F o. ( 1st |` ( _V X. _V ) ) ) ) = U_ x e. A ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) ) )

Distinct variable groups:   x, A   x, F

Proof of Theorem fparlem3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coiun 5523	. 2 |- ( `' ( 1st |` ( _V X. _V ) ) o. U_ x e. A ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) ) = U_ x e. A ( `' ( 1st |` ( _V X. _V ) ) o. ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
2		inss1 3714	. . . . 5 |- ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ dom F
3		fndm 5686	. . . . 5 |- ( F Fn A -> dom F = A )
4	2, 3	syl5sseq 3547	. . . 4 |- ( F Fn A -> ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ A )
5		dfco2a 5513	. . . 4 |- ( ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ A -> ( F o. ( 1st |` ( _V X. _V ) ) ) = U_ x e. A ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
6	4, 5	syl 16	. . 3 |- ( F Fn A -> ( F o. ( 1st |` ( _V X. _V ) ) ) = U_ x e. A ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
7	6	coeq2d 5175	. 2 |- ( F Fn A -> ( `' ( 1st |` ( _V X. _V ) ) o. ( F o. ( 1st |` ( _V X. _V ) ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. U_ x e. A ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) ) )
8		inss1 3714	. . . . . . . . 9 |- ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ dom ( { ( F ` x ) } X. ( { x } X. _V ) )
9		dmxpss 5445	. . . . . . . . 9 |- dom ( { ( F ` x ) } X. ( { x } X. _V ) ) C_ { ( F ` x ) }
10	8, 9	sstri 3508	. . . . . . . 8 |- ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ { ( F ` x ) }
11		dfco2a 5513	. . . . . . . 8 |- ( ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ { ( F ` x ) } -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = U_ y e. { ( F ` x ) } ( ( `' ( 1st |` ( _V X. _V ) ) " { y } ) X. ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) ) )
12	10, 11	ax-mp 5	. . . . . . 7 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = U_ y e. { ( F ` x ) } ( ( `' ( 1st |` ( _V X. _V ) ) " { y } ) X. ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) )
13		fvex 5882	. . . . . . . 8 |- ( F ` x ) e. _V
14		fparlem1 6899	. . . . . . . . . 10 |- ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { y } X. _V )
15		sneq 4042	. . . . . . . . . . 11 |- ( y = ( F ` x ) -> { y } = { ( F ` x ) } )
16	15	xpeq1d 5031	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( { y } X. _V ) = ( { ( F ` x ) } X. _V ) )
17	14, 16	syl5eq 2510	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { ( F ` x ) } X. _V ) )
18	15	imaeq2d 5347	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) )
19		df-ima 5021	. . . . . . . . . . 11 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } )
20		ssid 3518	. . . . . . . . . . . . . 14 |- { ( F ` x ) } C_ { ( F ` x ) }
21		xpssres 5318	. . . . . . . . . . . . . 14 |- ( { ( F ` x ) } C_ { ( F ` x ) } -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { ( F ` x ) } X. ( { x } X. _V ) ) )
22	20, 21	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { ( F ` x ) } X. ( { x } X. _V ) )
23	22	rneqi 5239	. . . . . . . . . . . 12 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ran ( { ( F ` x ) } X. ( { x } X. _V ) )
24	13	snnz 4150	. . . . . . . . . . . . 13 |- { ( F ` x ) } =/= (/)
25		rnxp 5444	. . . . . . . . . . . . 13 |- ( { ( F ` x ) } =/= (/) -> ran ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( { x } X. _V ) )
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 |- ran ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( { x } X. _V )
27	23, 26	eqtri 2486	. . . . . . . . . . 11 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { x } X. _V )
28	19, 27	eqtri 2486	. . . . . . . . . 10 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ( { x } X. _V )
29	18, 28	syl6eq 2514	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( { x } X. _V ) )
30	17, 29	xpeq12d 5033	. . . . . . . 8 |- ( y = ( F ` x ) -> ( ( `' ( 1st |` ( _V X. _V ) ) " { y } ) X. ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) ) )
31	13, 30	iunxsn 4415	. . . . . . 7 |- U_ y e. { ( F ` x ) } ( ( `' ( 1st |` ( _V X. _V ) ) " { y } ) X. ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
32	12, 31	eqtri 2486	. . . . . 6 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
33	32	cnveqi 5187	. . . . 5 |- `' ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
34		cnvco 5198	. . . . 5 |- `' ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) )
35		cnvxp 5431	. . . . 5 |- `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
36	33, 34, 35	3eqtr3i 2494	. . . 4 |- ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
37		fparlem1 6899	. . . . . . . . 9 |- ( `' ( 1st |` ( _V X. _V ) ) " { x } ) = ( { x } X. _V )
38	37	xpeq2i 5029	. . . . . . . 8 |- ( { ( F ` x ) } X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( { ( F ` x ) } X. ( { x } X. _V ) )
39		fnsnfv 5933	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> { ( F ` x ) } = ( F " { x } ) )
40	39	xpeq1d 5031	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( { ( F ` x ) } X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
41	38, 40	syl5eqr 2512	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
42	41	cnveqd 5188	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
43		cnvxp 5431	. . . . . 6 |- `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) )
44	42, 43	syl6eq 2514	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
45	44	coeq2d 5175	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) ) )
46	36, 45	syl5eqr 2512	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) ) )
47	46	iuneq2dv 4354	. 2 |- ( F Fn A -> U_ x e. A ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) ) = U_ x e. A ( `' ( 1st |` ( _V X. _V ) ) o. ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) ) )
48	1, 7, 47	3eqtr4a 2524	1 |- ( F Fn A -> ( `' ( 1st |` ( _V X. _V ) ) o. ( F o. ( 1st |` ( _V X. _V ) ) ) ) = U_ x e. A ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   {csn 4032   U_ciun 4332   X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   |` cres 5010   "cima 5011   o. ccom 5012   Fn wfn 5589   ` cfv 5594   1stc1st 6797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-1st 6799  df-2nd 6800

This theorem is referenced by:  fpar  6903