Theorem imainrect 5575

Description: Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Assertion

Ref	Expression
imainrect	|- ( ( G i^i ( A X. B ) ) " Y ) = ( ( G " ( Y i^i A ) ) i^i B )

Proof of Theorem imainrect

Step	Hyp	Ref	Expression
1		df-res 5126	. . 3 |- ( ( G i^i ( A X. B ) ) |` Y ) = ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
2	1	rneqi 5352	. 2 |- ran ( ( G i^i ( A X. B ) ) |` Y ) = ran ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
3		df-ima 5127	. 2 |- ( ( G i^i ( A X. B ) ) " Y ) = ran ( ( G i^i ( A X. B ) ) |` Y )
4		df-ima 5127	. . . . 5 |- ( G " ( Y i^i A ) ) = ran ( G |` ( Y i^i A ) )
5		df-res 5126	. . . . . 6 |- ( G |` ( Y i^i A ) ) = ( G i^i ( ( Y i^i A ) X. _V ) )
6	5	rneqi 5352	. . . . 5 |- ran ( G |` ( Y i^i A ) ) = ran ( G i^i ( ( Y i^i A ) X. _V ) )
7	4, 6	eqtri 2644	. . . 4 |- ( G " ( Y i^i A ) ) = ran ( G i^i ( ( Y i^i A ) X. _V ) )
8	7	ineq1i 3810	. . 3 |- ( ( G " ( Y i^i A ) ) i^i B ) = ( ran ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B )
9		cnvin 5540	. . . . . 6 |- `' ( ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( _V X. B ) ) = ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i `' ( _V X. B ) )
10		inxp 5254	. . . . . . . . . 10 |- ( ( A X. _V ) i^i ( _V X. B ) ) = ( ( A i^i _V ) X. ( _V i^i B ) )
11		inv1 3970	. . . . . . . . . . 11 |- ( A i^i _V ) = A
12		incom 3805	. . . . . . . . . . . 12 |- ( _V i^i B ) = ( B i^i _V )
13		inv1 3970	. . . . . . . . . . . 12 |- ( B i^i _V ) = B
14	12, 13	eqtri 2644	. . . . . . . . . . 11 |- ( _V i^i B ) = B
15	11, 14	xpeq12i 5137	. . . . . . . . . 10 |- ( ( A i^i _V ) X. ( _V i^i B ) ) = ( A X. B )
16	10, 15	eqtr2i 2645	. . . . . . . . 9 |- ( A X. B ) = ( ( A X. _V ) i^i ( _V X. B ) )
17	16	ineq2i 3811	. . . . . . . 8 |- ( ( G i^i ( Y X. _V ) ) i^i ( A X. B ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( ( A X. _V ) i^i ( _V X. B ) ) )
18		in32 3825	. . . . . . . 8 |- ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( A X. B ) )
19		xpindir 5256	. . . . . . . . . . . 12 |- ( ( Y i^i A ) X. _V ) = ( ( Y X. _V ) i^i ( A X. _V ) )
20	19	ineq2i 3811	. . . . . . . . . . 11 |- ( G i^i ( ( Y i^i A ) X. _V ) ) = ( G i^i ( ( Y X. _V ) i^i ( A X. _V ) ) )
21		inass 3823	. . . . . . . . . . 11 |- ( ( G i^i ( Y X. _V ) ) i^i ( A X. _V ) ) = ( G i^i ( ( Y X. _V ) i^i ( A X. _V ) ) )
22	20, 21	eqtr4i 2647	. . . . . . . . . 10 |- ( G i^i ( ( Y i^i A ) X. _V ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( A X. _V ) )
23	22	ineq1i 3810	. . . . . . . . 9 |- ( ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( _V X. B ) ) = ( ( ( G i^i ( Y X. _V ) ) i^i ( A X. _V ) ) i^i ( _V X. B ) )
24		inass 3823	. . . . . . . . 9 |- ( ( ( G i^i ( Y X. _V ) ) i^i ( A X. _V ) ) i^i ( _V X. B ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( ( A X. _V ) i^i ( _V X. B ) ) )
25	23, 24	eqtri 2644	. . . . . . . 8 |- ( ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( _V X. B ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( ( A X. _V ) i^i ( _V X. B ) ) )
26	17, 18, 25	3eqtr4i 2654	. . . . . . 7 |- ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) ) = ( ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( _V X. B ) )
27	26	cnveqi 5297	. . . . . 6 |- `' ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) ) = `' ( ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( _V X. B ) )
28		df-res 5126	. . . . . . 7 |- ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) |` B ) = ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( B X. _V ) )
29		cnvxp 5551	. . . . . . . 8 |- `' ( _V X. B ) = ( B X. _V )
30	29	ineq2i 3811	. . . . . . 7 |- ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i `' ( _V X. B ) ) = ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i ( B X. _V ) )
31	28, 30	eqtr4i 2647	. . . . . 6 |- ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) |` B ) = ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i `' ( _V X. B ) )
32	9, 27, 31	3eqtr4ri 2655	. . . . 5 |- ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) |` B ) = `' ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
33	32	dmeqi 5325	. . . 4 |- dom ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) |` B ) = dom `' ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
34		incom 3805	. . . . 5 |- ( B i^i dom `' ( G i^i ( ( Y i^i A ) X. _V ) ) ) = ( dom `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B )
35		dmres 5419	. . . . 5 |- dom ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) |` B ) = ( B i^i dom `' ( G i^i ( ( Y i^i A ) X. _V ) ) )
36		df-rn 5125	. . . . . 6 |- ran ( G i^i ( ( Y i^i A ) X. _V ) ) = dom `' ( G i^i ( ( Y i^i A ) X. _V ) )
37	36	ineq1i 3810	. . . . 5 |- ( ran ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B ) = ( dom `' ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B )
38	34, 35, 37	3eqtr4ri 2655	. . . 4 |- ( ran ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B ) = dom ( `' ( G i^i ( ( Y i^i A ) X. _V ) ) |` B )
39		df-rn 5125	. . . 4 |- ran ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) ) = dom `' ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
40	33, 38, 39	3eqtr4ri 2655	. . 3 |- ran ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) ) = ( ran ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B )
41	8, 40	eqtr4i 2647	. 2 |- ( ( G " ( Y i^i A ) ) i^i B ) = ran ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
42	2, 3, 41	3eqtr4i 2654	1 |- ( ( G i^i ( A X. B ) ) " Y ) = ( ( G " ( Y i^i A ) ) i^i B )

Syntax hints:   = wceq 1483   _Vcvv 3200   i^i cin 3573   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  ecinxp  7822  marypha1lem  8339