Theorem fparlem3 6695

Description: Lemma for fpar 6697. (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 5368	. 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 3591	. . . . 5 |- ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ dom F
3		fndm 5531	. . . . 5 |- ( F Fn A -> dom F = A )
4	2, 3	syl5sseq 3425	. . . 4 |- ( F Fn A -> ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ A )
5		dfco2a 5359	. . . 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 5023	. 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 3591	. . . . . . . . 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 5290	. . . . . . . . 9 |- dom ( { ( F ` x ) } X. ( { x } X. _V ) ) C_ { ( F ` x ) }
10	8, 9	sstri 3386	. . . . . . . 8 |- ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ { ( F ` x ) }
11		dfco2a 5359	. . . . . . . 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 5722	. . . . . . . 8 |- ( F ` x ) e. _V
14		fparlem1 6693	. . . . . . . . . 10 |- ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { y } X. _V )
15		sneq 3908	. . . . . . . . . . 11 |- ( y = ( F ` x ) -> { y } = { ( F ` x ) } )
16	15	xpeq1d 4884	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( { y } X. _V ) = ( { ( F ` x ) } X. _V ) )
17	14, 16	syl5eq 2487	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { ( F ` x ) } X. _V ) )
18	15	imaeq2d 5190	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) )
19		df-ima 4874	. . . . . . . . . . 11 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } )
20		ssid 3396	. . . . . . . . . . . . . 14 |- { ( F ` x ) } C_ { ( F ` x ) }
21		xpssres 5165	. . . . . . . . . . . . . 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 5087	. . . . . . . . . . . 12 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ran ( { ( F ` x ) } X. ( { x } X. _V ) )
24	13	snnz 4014	. . . . . . . . . . . . 13 |- { ( F ` x ) } =/= (/)
25		rnxp 5289	. . . . . . . . . . . . 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 2463	. . . . . . . . . . 11 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { x } X. _V )
28	19, 27	eqtri 2463	. . . . . . . . . 10 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ( { x } X. _V )
29	18, 28	syl6eq 2491	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( { x } X. _V ) )
30	17, 29	xpeq12d 4886	. . . . . . . 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 4271	. . . . . . 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 2463	. . . . . 6 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
33	32	cnveqi 5035	. . . . 5 |- `' ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
34		cnvco 5046	. . . . 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 5276	. . . . 5 |- `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
36	33, 34, 35	3eqtr3i 2471	. . . 4 |- ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
37		fparlem1 6693	. . . . . . . . 9 |- ( `' ( 1st |` ( _V X. _V ) ) " { x } ) = ( { x } X. _V )
38	37	xpeq2i 4882	. . . . . . . 8 |- ( { ( F ` x ) } X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( { ( F ` x ) } X. ( { x } X. _V ) )
39		fnsnfv 5772	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> { ( F ` x ) } = ( F " { x } ) )
40	39	xpeq1d 4884	. . . . . . . 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 2489	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
42	41	cnveqd 5036	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
43		cnvxp 5276	. . . . . 6 |- `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) )
44	42, 43	syl6eq 2491	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
45	44	coeq2d 5023	. . . 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 2489	. . 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 4213	. 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 2501	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 1369   e. wcel 1756   =/= wne 2620   _Vcvv 2993   i^i cin 3348   C_ wss 3349   (/)c0 3658   {csn 3898   U_ciun 4192   X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862   |` cres 4863   "cima 4864   o. ccom 4865   Fn wfn 5434   ` cfv 5439   1stc1st 6596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-1st 6598  df-2nd 6599

This theorem is referenced by:  fpar  6697