Theorem fparlem3 6917

Description: Lemma for fpar 6919. (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 5352	. 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 3643	. . . . 5 |- ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ dom F
3		fndm 5685	. . . . 5 |- ( F Fn A -> dom F = A )
4	2, 3	syl5sseq 3466	. . . 4 |- ( F Fn A -> ( dom F i^i ran ( 1st |` ( _V X. _V ) ) ) C_ A )
5		dfco2a 5342	. . . 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 5002	. 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 3643	. . . . . . . . 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 5274	. . . . . . . . 9 |- dom ( { ( F ` x ) } X. ( { x } X. _V ) ) C_ { ( F ` x ) }
10	8, 9	sstri 3427	. . . . . . . 8 |- ( dom ( { ( F ` x ) } X. ( { x } X. _V ) ) i^i ran ( 1st |` ( _V X. _V ) ) ) C_ { ( F ` x ) }
11		dfco2a 5342	. . . . . . . 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 5889	. . . . . . . 8 |- ( F ` x ) e. _V
14		fparlem1 6915	. . . . . . . . . 10 |- ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { y } X. _V )
15		sneq 3969	. . . . . . . . . . 11 |- ( y = ( F ` x ) -> { y } = { ( F ` x ) } )
16	15	xpeq1d 4862	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( { y } X. _V ) = ( { ( F ` x ) } X. _V ) )
17	14, 16	syl5eq 2517	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( `' ( 1st |` ( _V X. _V ) ) " { y } ) = ( { ( F ` x ) } X. _V ) )
18	15	imaeq2d 5174	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) )
19		df-ima 4852	. . . . . . . . . . 11 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } )
20		ssid 3437	. . . . . . . . . . . . . 14 |- { ( F ` x ) } C_ { ( F ` x ) }
21		xpssres 5145	. . . . . . . . . . . . . 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 5067	. . . . . . . . . . . 12 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ran ( { ( F ` x ) } X. ( { x } X. _V ) )
24	13	snnz 4081	. . . . . . . . . . . . 13 |- { ( F ` x ) } =/= (/)
25		rnxp 5273	. . . . . . . . . . . . 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 2493	. . . . . . . . . . 11 |- ran ( ( { ( F ` x ) } X. ( { x } X. _V ) ) |` { ( F ` x ) } ) = ( { x } X. _V )
28	19, 27	eqtri 2493	. . . . . . . . . 10 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { ( F ` x ) } ) = ( { x } X. _V )
29	18, 28	syl6eq 2521	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( { ( F ` x ) } X. ( { x } X. _V ) ) " { y } ) = ( { x } X. _V ) )
30	17, 29	xpeq12d 4864	. . . . . . . 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 4352	. . . . . . 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 2493	. . . . . 6 |- ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
33	32	cnveqi 5014	. . . . 5 |- `' ( ( { ( F ` x ) } X. ( { x } X. _V ) ) o. ( 1st |` ( _V X. _V ) ) ) = `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) )
34		cnvco 5025	. . . . 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 5260	. . . . 5 |- `' ( ( { ( F ` x ) } X. _V ) X. ( { x } X. _V ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
36	33, 34, 35	3eqtr3i 2501	. . . 4 |- ( `' ( 1st |` ( _V X. _V ) ) o. `' ( { ( F ` x ) } X. ( { x } X. _V ) ) ) = ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) )
37		fparlem1 6915	. . . . . . . . 9 |- ( `' ( 1st |` ( _V X. _V ) ) " { x } ) = ( { x } X. _V )
38	37	xpeq2i 4860	. . . . . . . 8 |- ( { ( F ` x ) } X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( { ( F ` x ) } X. ( { x } X. _V ) )
39		fnsnfv 5940	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> { ( F ` x ) } = ( F " { x } ) )
40	39	xpeq1d 4862	. . . . . . . 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 2519	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
42	41	cnveqd 5015	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) )
43		cnvxp 5260	. . . . . 6 |- `' ( ( F " { x } ) X. ( `' ( 1st |` ( _V X. _V ) ) " { x } ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) )
44	42, 43	syl6eq 2521	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> `' ( { ( F ` x ) } X. ( { x } X. _V ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) " { x } ) X. ( F " { x } ) ) )
45	44	coeq2d 5002	. . . 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 2519	. . 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 4291	. 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 2531	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 376   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   {csn 3959   U_ciun 4269   X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   o. ccom 4843   Fn wfn 5584   ` cfv 5589   1stc1st 6810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-1st 6812  df-2nd 6813

This theorem is referenced by:  fpar  6919