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Theorem imainrect 5358
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
StepHypRef Expression
1 df-res 4925 . . 3  |-  ( ( G  i^i  ( A  X.  B ) )  |`  Y )  =  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
21rneqi 5142 . 2  |-  ran  (
( G  i^i  ( A  X.  B ) )  |`  Y )  =  ran  ( ( G  i^i  ( A  X.  B
) )  i^i  ( Y  X.  _V ) )
3 df-ima 4926 . 2  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ran  ( ( G  i^i  ( A  X.  B ) )  |`  Y )
4 df-ima 4926 . . . . 5  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  |`  ( Y  i^i  A ) )
5 df-res 4925 . . . . . 6  |-  ( G  |`  ( Y  i^i  A
) )  =  ( G  i^i  ( ( Y  i^i  A )  X.  _V ) )
65rneqi 5142 . . . . 5  |-  ran  ( G  |`  ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
74, 6eqtri 2411 . . . 4  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
87ineq1i 3610 . . 3  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ( ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  i^i 
B )
9 cnvin 5323 . . . . . 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 5048 . . . . . . . . . 10  |-  ( ( A  X.  _V )  i^i  ( _V  X.  B
) )  =  ( ( A  i^i  _V )  X.  ( _V  i^i  B ) )
11 inv1 3739 . . . . . . . . . . 11  |-  ( A  i^i  _V )  =  A
12 incom 3605 . . . . . . . . . . . 12  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
13 inv1 3739 . . . . . . . . . . . 12  |-  ( B  i^i  _V )  =  B
1412, 13eqtri 2411 . . . . . . . . . . 11  |-  ( _V 
i^i  B )  =  B
1511, 14xpeq12i 4935 . . . . . . . . . 10  |-  ( ( A  i^i  _V )  X.  ( _V  i^i  B
) )  =  ( A  X.  B )
1610, 15eqtr2i 2412 . . . . . . . . 9  |-  ( A  X.  B )  =  ( ( A  X.  _V )  i^i  ( _V  X.  B ) )
1716ineq2i 3611 . . . . . . . 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 3624 . . . . . . . 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 5050 . . . . . . . . . . . 12  |-  ( ( Y  i^i  A )  X.  _V )  =  ( ( Y  X.  _V )  i^i  ( A  X.  _V ) )
2019ineq2i 3611 . . . . . . . . . . 11  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( G  i^i  (
( Y  X.  _V )  i^i  ( A  X.  _V ) ) )
21 inass 3622 . . . . . . . . . . 11  |-  ( ( G  i^i  ( Y  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( G  i^i  ( ( Y  X.  _V )  i^i  ( A  X.  _V ) ) )
2220, 21eqtr4i 2414 . . . . . . . . . 10  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( ( G  i^i  ( Y  X.  _V )
)  i^i  ( A  X.  _V ) )
2322ineq1i 3610 . . . . . . . . 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 3622 . . . . . . . . 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 ) ) )
2523, 24eqtri 2411 . . . . . . . 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
) ) )
2617, 18, 253eqtr4i 2421 . . . . . . 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 ) )
2726cnveqi 5090 . . . . . 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 4925 . . . . . . 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 5334 . . . . . . . 8  |-  `' ( _V  X.  B )  =  ( B  X.  _V )
3029ineq2i 3611 . . . . . . 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 ) )
3128, 30eqtr4i 2414 . . . . . 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 ) )
329, 27, 313eqtr4ri 2422 . . . . 5  |-  ( `' ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)  |`  B )  =  `' ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
3332dmeqi 5117 . . . 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 3605 . . . . 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 5206 . . . . 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 4924 . . . . . 6  |-  ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  dom  `' ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )
3736ineq1i 3610 . . . . 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 )
3834, 35, 373eqtr4ri 2422 . . . 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 4924 . . . 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 ) )
4033, 38, 393eqtr4ri 2422 . . 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 )
418, 40eqtr4i 2414 . 2  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ran  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
422, 3, 413eqtr4i 2421 1  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ( ( G "
( Y  i^i  A
) )  i^i  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399   _Vcvv 3034    i^i cin 3388    X. cxp 4911   `'ccnv 4912   dom cdm 4913   ran crn 4914    |` cres 4915   "cima 4916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
This theorem is referenced by:  ecinxp  7304  marypha1lem  7808
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