Theorem imainrect 5208

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 4761	. . 3 |- ( ( G i^i ( A X. B ) ) |` Y ) = ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
2	1	rneqi 4985	. 2 |- ran ( ( G i^i ( A X. B ) ) |` Y ) = ran ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
3		df-ima 4762	. 2 |- ( ( G i^i ( A X. B ) ) " Y ) = ran ( ( G i^i ( A X. B ) ) |` Y )
4		df-ima 4762	. . . . 5 |- ( G " ( Y i^i A ) ) = ran ( G |` ( Y i^i A ) )
5		df-res 4761	. . . . . 6 |- ( G |` ( Y i^i A ) ) = ( G i^i ( ( Y i^i A ) X. _V ) )
6	5	rneqi 4985	. . . . 5 |- ran ( G |` ( Y i^i A ) ) = ran ( G i^i ( ( Y i^i A ) X. _V ) )
7	4, 6	eqtri 2253	. . . 4 |- ( G " ( Y i^i A ) ) = ran ( G i^i ( ( Y i^i A ) X. _V ) )
8	7	ineq1i 3418	. . 3 |- ( ( G " ( Y i^i A ) ) i^i B ) = ( ran ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B )
9		cnvin 5170	. . . . . 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 4889	. . . . . . . . . 10 |- ( ( A X. _V ) i^i ( _V X. B ) ) = ( ( A i^i _V ) X. ( _V i^i B ) )
11		inv1 3545	. . . . . . . . . . 11 |- ( A i^i _V ) = A
12		incom 3411	. . . . . . . . . . . 12 |- ( _V i^i B ) = ( B i^i _V )
13		inv1 3545	. . . . . . . . . . . 12 |- ( B i^i _V ) = B
14	12, 13	eqtri 2253	. . . . . . . . . . 11 |- ( _V i^i B ) = B
15	11, 14	xpeq12i 4771	. . . . . . . . . 10 |- ( ( A i^i _V ) X. ( _V i^i B ) ) = ( A X. B )
16	10, 15	eqtr2i 2254	. . . . . . . . 9 |- ( A X. B ) = ( ( A X. _V ) i^i ( _V X. B ) )
17	16	ineq2i 3419	. . . . . . . 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 3433	. . . . . . . 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 4891	. . . . . . . . . . . 12 |- ( ( Y i^i A ) X. _V ) = ( ( Y X. _V ) i^i ( A X. _V ) )
20	19	ineq2i 3419	. . . . . . . . . . 11 |- ( G i^i ( ( Y i^i A ) X. _V ) ) = ( G i^i ( ( Y X. _V ) i^i ( A X. _V ) ) )
21		inass 3431	. . . . . . . . . . 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 2256	. . . . . . . . . 10 |- ( G i^i ( ( Y i^i A ) X. _V ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( A X. _V ) )
23	22	ineq1i 3418	. . . . . . . . 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 3431	. . . . . . . . 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 2253	. . . . . . . 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 2263	. . . . . . 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 4930	. . . . . 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 4761	. . . . . . 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 5181	. . . . . . . 8 |- `' ( _V X. B ) = ( B X. _V )
30	29	ineq2i 3419	. . . . . . 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 2256	. . . . . 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 2264	. . . . 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 4957	. . . 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 3411	. . . . 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 5059	. . . . 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 4760	. . . . . 6 |- ran ( G i^i ( ( Y i^i A ) X. _V ) ) = dom `' ( G i^i ( ( Y i^i A ) X. _V ) )
37	36	ineq1i 3418	. . . . 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 2264	. . . 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 4760	. . . 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 2264	. . 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 2256	. 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 2263	1 |- ( ( G i^i ( A X. B ) ) " Y ) = ( ( G " ( Y i^i A ) ) i^i B )

Syntax hints:   = wceq 1398   _Vcvv 2813   i^i cin 3210   X. cxp 4747   `'ccnv 4748   dom cdm 4749   ran crn 4750   |` cres 4751   "cima 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  ecinxp  6844