Theorem fparlem3 6898

Description: Lemma for fpar 6900. (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 5345	. 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 3652	. . . . 5 |- ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ dom F
3		fndm 5675	. . . . 5 |- ( F Fn A -> dom F = A )
4	2, 3	syl5sseq 3480	. . . 4 |- ( F Fn A -> ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ A )
5		dfco2a 5335	. . . 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 17	. . 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 4997	. 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 3652	. . . . . . . . 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 5268	. . . . . . . . 9 |- dom ( { ( F ` x ) } X. ( { x } X. _V ) ) C_ { ( F ` x ) }
10	8, 9	sstri 3441	. . . . . . . 8 |- ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ { ( F ` x ) }
11		dfco2a 5335	. . . . . . . 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 5875	. . . . . . . 8 |- ( F ` x ) e. _V
14		fparlem1 6896	. . . . . . . . . 10 |- ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { y } X. _V )
15		sneq 3978	. . . . . . . . . . 11 |- ( y = ( F ` x ) -> { y } = { ( F ` x ) } )
16	15	xpeq1d 4857	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( { y } X. _V ) = ( { ( F ` x ) } X. _V ) )
17	14, 16	syl5eq 2497	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { ( F ` x ) } X. _V ) )
18	15	imaeq2d 5168	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) )
19		df-ima 4847	. . . . . . . . . . 11 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } )
20		ssid 3451	. . . . . . . . . . . . . 14 |- { ( F ` x ) } C_ { ( F ` x ) }
21		xpssres 5139	. . . . . . . . . . . . . 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 5061	. . . . . . . . . . . 12 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ran ( { ( F ` x ) } X. ( { x } X. _V ) )
24	13	snnz 4090	. . . . . . . . . . . . 13 |- { ( F ` x ) } =/= (/)
25		rnxp 5267	. . . . . . . . . . . . 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 2473	. . . . . . . . . . 11 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { x } X. _V )
28	19, 27	eqtri 2473	. . . . . . . . . 10 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ( { x } X. _V )
29	18, 28	syl6eq 2501	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( { x } X. _V ) )
30	17, 29	xpeq12d 4859	. . . . . . . 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 4361	. . . . . . 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 2473	. . . . . 6 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
33	32	cnveqi 5009	. . . . 5 |- `' ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
34		cnvco 5020	. . . . 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 5254	. . . . 5 |- `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
36	33, 34, 35	3eqtr3i 2481	. . . 4 |- ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
37		fparlem1 6896	. . . . . . . . 9 |- ( `' ( 1st |` ( _V X. _V ) ) " { x } ) = ( { x } X. _V )
38	37	xpeq2i 4855	. . . . . . . 8 |- ( { ( F ` x ) } X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( { ( F ` x ) } X. ( { x } X. _V ) )
39		fnsnfv 5925	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> { ( F ` x ) } = ( F " { x } ) )
40	39	xpeq1d 4857	. . . . . . . 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 2499	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
42	41	cnveqd 5010	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
43		cnvxp 5254	. . . . . 6 |- `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) )
44	42, 43	syl6eq 2501	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
45	44	coeq2d 4997	. . . 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 2499	. . 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 4300	. 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 2511	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 371   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   U_ciun 4278   X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   |` cres 4836   "cima 4837   o. ccom 4838   Fn wfn 5577   ` cfv 5582   1stc1st 6791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-1st 6793  df-2nd 6794

This theorem is referenced by:  fpar  6900