Theorem fparlem4 7280

Description: Lemma for fpar 7281. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fparlem4	|- ( G Fn B -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) = U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) )

Distinct variable groups:   y, B   y, G

Proof of Theorem fparlem4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coiun 5645	. 2 |- ( `' ( 2nd |` ( _V X. _V ) ) o. U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) = U_ y e. B ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
2		inss1 3833	. . . . 5 |- ( dom G i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ dom G
3		fndm 5990	. . . . 5 |- ( G Fn B -> dom G = B )
4	2, 3	syl5sseq 3653	. . . 4 |- ( G Fn B -> ( dom G i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ B )
5		dfco2a 5635	. . . 4 |- ( ( dom G i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ B -> ( G o. ( 2nd |` ( _V X. _V ) ) ) = U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
6	4, 5	syl 17	. . 3 |- ( G Fn B -> ( G o. ( 2nd |` ( _V X. _V ) ) ) = U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
7	6	coeq2d 5284	. 2 |- ( G Fn B -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
8		inss1 3833	. . . . . . . . 9 |- ( dom ( { ( G ` y ) } X. ( _V X. { y } ) ) i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ dom ( { ( G ` y ) } X. ( _V X. { y } ) )
9		dmxpss 5565	. . . . . . . . 9 |- dom ( { ( G ` y ) } X. ( _V X. { y } ) ) C_ { ( G ` y ) }
10	8, 9	sstri 3612	. . . . . . . 8 |- ( dom ( { ( G ` y ) } X. ( _V X. { y } ) ) i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ { ( G ` y ) }
11		dfco2a 5635	. . . . . . . 8 |- ( ( dom ( { ( G ` y ) } X. ( _V X. { y } ) ) i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ { ( G ` y ) } -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = U_ x e. { ( G ` y ) } ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) ) )
12	10, 11	ax-mp 5	. . . . . . 7 |- ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = U_ x e. { ( G ` y ) } ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) )
13		fvex 6201	. . . . . . . 8 |- ( G ` y ) e. _V
14		fparlem2 7278	. . . . . . . . . 10 |- ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) = ( _V X. { x } )
15		sneq 4187	. . . . . . . . . . 11 |- ( x = ( G ` y ) -> { x } = { ( G ` y ) } )
16	15	xpeq2d 5139	. . . . . . . . . 10 |- ( x = ( G ` y ) -> ( _V X. { x } ) = ( _V X. { ( G ` y ) } ) )
17	14, 16	syl5eq 2668	. . . . . . . . 9 |- ( x = ( G ` y ) -> ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) = ( _V X. { ( G ` y ) } ) )
18	15	imaeq2d 5466	. . . . . . . . . 10 |- ( x = ( G ` y ) -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) = ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { ( G ` y ) } ) )
19		df-ima 5127	. . . . . . . . . . 11 |- ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { ( G ` y ) } ) = ran ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } )
20		ssid 3624	. . . . . . . . . . . . . 14 |- { ( G ` y ) } C_ { ( G ` y ) }
21		xpssres 5434	. . . . . . . . . . . . . 14 |- ( { ( G ` y ) } C_ { ( G ` y ) } -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ( { ( G ` y ) } X. ( _V X. { y } ) ) )
22	20, 21	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ( { ( G ` y ) } X. ( _V X. { y } ) )
23	22	rneqi 5352	. . . . . . . . . . . 12 |- ran ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ran ( { ( G ` y ) } X. ( _V X. { y } ) )
24	13	snnz 4309	. . . . . . . . . . . . 13 |- { ( G ` y ) } =/= (/)
25		rnxp 5564	. . . . . . . . . . . . 13 |- ( { ( G ` y ) } =/= (/) -> ran ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( _V X. { y } ) )
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 |- ran ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( _V X. { y } )
27	23, 26	eqtri 2644	. . . . . . . . . . 11 |- ran ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ( _V X. { y } )
28	19, 27	eqtri 2644	. . . . . . . . . 10 |- ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { ( G ` y ) } ) = ( _V X. { y } )
29	18, 28	syl6eq 2672	. . . . . . . . 9 |- ( x = ( G ` y ) -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) = ( _V X. { y } ) )
30	17, 29	xpeq12d 5140	. . . . . . . 8 |- ( x = ( G ` y ) -> ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) ) = ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) ) )
31	13, 30	iunxsn 4603	. . . . . . 7 |- U_ x e. { ( G ` y ) } ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) ) = ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) )
32	12, 31	eqtri 2644	. . . . . 6 |- ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) )
33	32	cnveqi 5297	. . . . 5 |- `' ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = `' ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) )
34		cnvco 5308	. . . . 5 |- `' ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. `' ( { ( G ` y ) } X. ( _V X. { y } ) ) )
35		cnvxp 5551	. . . . 5 |- `' ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) ) = ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) )
36	33, 34, 35	3eqtr3i 2652	. . . 4 |- ( `' ( 2nd |` ( _V X. _V ) ) o. `' ( { ( G ` y ) } X. ( _V X. { y } ) ) ) = ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) )
37		fparlem2 7278	. . . . . . . . 9 |- ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) = ( _V X. { y } )
38	37	xpeq2i 5136	. . . . . . . 8 |- ( { ( G ` y ) } X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) = ( { ( G ` y ) } X. ( _V X. { y } ) )
39		fnsnfv 6258	. . . . . . . . 9 |- ( ( G Fn B /\ y e. B ) -> { ( G ` y ) } = ( G " { y } ) )
40	39	xpeq1d 5138	. . . . . . . 8 |- ( ( G Fn B /\ y e. B ) -> ( { ( G ` y ) } X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) = ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) )
41	38, 40	syl5eqr 2670	. . . . . . 7 |- ( ( G Fn B /\ y e. B ) -> ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) )
42	41	cnveqd 5298	. . . . . 6 |- ( ( G Fn B /\ y e. B ) -> `' ( { ( G ` y ) } X. ( _V X. { y } ) ) = `' ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) )
43		cnvxp 5551	. . . . . 6 |- `' ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) = ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) )
44	42, 43	syl6eq 2672	. . . . 5 |- ( ( G Fn B /\ y e. B ) -> `' ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
45	44	coeq2d 5284	. . . 4 |- ( ( G Fn B /\ y e. B ) -> ( `' ( 2nd |` ( _V X. _V ) ) o. `' ( { ( G ` y ) } X. ( _V X. { y } ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
46	36, 45	syl5eqr 2670	. . 3 |- ( ( G Fn B /\ y e. B ) -> ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
47	46	iuneq2dv 4542	. 2 |- ( G Fn B -> U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) = U_ y e. B ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
48	1, 7, 47	3eqtr4a 2682	1 |- ( G Fn B -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) = U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fn wfn 5883   ` cfv 5888   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fpar  7281