Theorem imainrect 5228

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 4781	. . 3 |- ( ( G i^i ( A X. B ) ) |` Y ) = ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
2	1	rneqi 5005	. 2 |- ran ( ( G i^i ( A X. B ) ) |` Y ) = ran ( ( G i^i ( A X. B ) ) i^i ( Y X. _V ) )
3		df-ima 4782	. 2 |- ( ( G i^i ( A X. B ) ) " Y ) = ran ( ( G i^i ( A X. B ) ) |` Y )
4		df-ima 4782	. . . . 5 |- ( G " ( Y i^i A ) ) = ran ( G |` ( Y i^i A ) )
5		df-res 4781	. . . . . 6 |- ( G |` ( Y i^i A ) ) = ( G i^i ( ( Y i^i A ) X. _V ) )
6	5	rneqi 5005	. . . . 5 |- ran ( G |` ( Y i^i A ) ) = ran ( G i^i ( ( Y i^i A ) X. _V ) )
7	4, 6	eqtri 2259	. . . 4 |- ( G " ( Y i^i A ) ) = ran ( G i^i ( ( Y i^i A ) X. _V ) )
8	7	ineq1i 3428	. . 3 |- ( ( G " ( Y i^i A ) ) i^i B ) = ( ran ( G i^i ( ( Y i^i A ) X. _V ) ) i^i B )
9		cnvin 5190	. . . . . 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 4909	. . . . . . . . . 10 |- ( ( A X. _V ) i^i ( _V X. B ) ) = ( ( A i^i _V ) X. ( _V i^i B ) )
11		inv1 3559	. . . . . . . . . . 11 |- ( A i^i _V ) = A
12		incom 3421	. . . . . . . . . . . 12 |- ( _V i^i B ) = ( B i^i _V )
13		inv1 3559	. . . . . . . . . . . 12 |- ( B i^i _V ) = B
14	12, 13	eqtri 2259	. . . . . . . . . . 11 |- ( _V i^i B ) = B
15	11, 14	xpeq12i 4791	. . . . . . . . . 10 |- ( ( A i^i _V ) X. ( _V i^i B ) ) = ( A X. B )
16	10, 15	eqtr2i 2260	. . . . . . . . 9 |- ( A X. B ) = ( ( A X. _V ) i^i ( _V X. B ) )
17	16	ineq2i 3429	. . . . . . . 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 3443	. . . . . . . 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 4911	. . . . . . . . . . . 12 |- ( ( Y i^i A ) X. _V ) = ( ( Y X. _V ) i^i ( A X. _V ) )
20	19	ineq2i 3429	. . . . . . . . . . 11 |- ( G i^i ( ( Y i^i A ) X. _V ) ) = ( G i^i ( ( Y X. _V ) i^i ( A X. _V ) ) )
21		inass 3441	. . . . . . . . . . 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 2262	. . . . . . . . . 10 |- ( G i^i ( ( Y i^i A ) X. _V ) ) = ( ( G i^i ( Y X. _V ) ) i^i ( A X. _V ) )
23	22	ineq1i 3428	. . . . . . . . 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 3441	. . . . . . . . 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 2259	. . . . . . . 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 2269	. . . . . . 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 4950	. . . . . 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 4781	. . . . . . 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 5201	. . . . . . . 8 |- `' ( _V X. B ) = ( B X. _V )
30	29	ineq2i 3429	. . . . . . 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 2262	. . . . . 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 2270	. . . . 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 4977	. . . 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 3421	. . . . 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 5079	. . . . 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 4780	. . . . . 6 |- ran ( G i^i ( ( Y i^i A ) X. _V ) ) = dom `' ( G i^i ( ( Y i^i A ) X. _V ) )
37	36	ineq1i 3428	. . . . 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 2270	. . . 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 4780	. . . 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 2270	. . 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 2262	. 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 2269	1 |- ( ( G i^i ( A X. B ) ) " Y ) = ( ( G " ( Y i^i A ) ) i^i B )

Syntax hints:   = wceq 1402   _Vcvv 2821   i^i cin 3219   X. cxp 4767   `'ccnv 4768   dom cdm 4769   ran crn 4770   |` cres 4771   "cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  ecinxp  6874