Theorem fparlem3 6885

Description: Lemma for fpar 6887. (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 5517	. 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 3718	. . . . 5 |- ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ dom F
3		fndm 5680	. . . . 5 |- ( F Fn A -> dom F = A )
4	2, 3	syl5sseq 3552	. . . 4 |- ( F Fn A -> ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ A )
5		dfco2a 5507	. . . 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 5165	. 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 3718	. . . . . . . . 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 5438	. . . . . . . . 9 |- dom ( { ( F ` x ) } X. ( { x } X. _V ) ) C_ { ( F ` x ) }
10	8, 9	sstri 3513	. . . . . . . 8 |- ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ { ( F ` x ) }
11		dfco2a 5507	. . . . . . . 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 5876	. . . . . . . 8 |- ( F ` x ) e. _V
14		fparlem1 6883	. . . . . . . . . 10 |- ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { y } X. _V )
15		sneq 4037	. . . . . . . . . . 11 |- ( y = ( F ` x ) -> { y } = { ( F ` x ) } )
16	15	xpeq1d 5022	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( { y } X. _V ) = ( { ( F ` x ) } X. _V ) )
17	14, 16	syl5eq 2520	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { ( F ` x ) } X. _V ) )
18	15	imaeq2d 5337	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) )
19		df-ima 5012	. . . . . . . . . . 11 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } )
20		ssid 3523	. . . . . . . . . . . . . 14 |- { ( F ` x ) } C_ { ( F ` x ) }
21		xpssres 5308	. . . . . . . . . . . . . 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 5229	. . . . . . . . . . . 12 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ran ( { ( F ` x ) } X. ( { x } X. _V ) )
24	13	snnz 4145	. . . . . . . . . . . . 13 |- { ( F ` x ) } =/= (/)
25		rnxp 5437	. . . . . . . . . . . . 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 2496	. . . . . . . . . . 11 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { x } X. _V )
28	19, 27	eqtri 2496	. . . . . . . . . 10 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ( { x } X. _V )
29	18, 28	syl6eq 2524	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( { x } X. _V ) )
30	17, 29	xpeq12d 5024	. . . . . . . 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 4405	. . . . . . 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 2496	. . . . . 6 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
33	32	cnveqi 5177	. . . . 5 |- `' ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
34		cnvco 5188	. . . . 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 5424	. . . . 5 |- `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
36	33, 34, 35	3eqtr3i 2504	. . . 4 |- ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
37		fparlem1 6883	. . . . . . . . 9 |- ( `' ( 1st |` ( _V X. _V ) ) " { x } ) = ( { x } X. _V )
38	37	xpeq2i 5020	. . . . . . . 8 |- ( { ( F ` x ) } X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( { ( F ` x ) } X. ( { x } X. _V ) )
39		fnsnfv 5927	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> { ( F ` x ) } = ( F " { x } ) )
40	39	xpeq1d 5022	. . . . . . . 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 2522	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
42	41	cnveqd 5178	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
43		cnvxp 5424	. . . . . 6 |- `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) )
44	42, 43	syl6eq 2524	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
45	44	coeq2d 5165	. . . 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 2522	. . 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 4347	. 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 2534	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 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3113   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   U_ciun 4325   X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   o. ccom 5003   Fn wfn 5583   ` cfv 5588   1stc1st 6782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-1st 6784  df-2nd 6785

This theorem is referenced by:  fpar  6887