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Theorem imainrect 5315
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 4893 . . 3  |-  ( ( G  i^i  ( A  X.  B ) )  |`  Y )  =  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
21rneqi 5099 . 2  |-  ran  (
( G  i^i  ( A  X.  B ) )  |`  Y )  =  ran  ( ( G  i^i  ( A  X.  B
) )  i^i  ( Y  X.  _V ) )
3 df-ima 4894 . 2  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ran  ( ( G  i^i  ( A  X.  B ) )  |`  Y )
4 df-ima 4894 . . . . 5  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  |`  ( Y  i^i  A ) )
5 df-res 4893 . . . . . 6  |-  ( G  |`  ( Y  i^i  A
) )  =  ( G  i^i  ( ( Y  i^i  A )  X.  _V ) )
65rneqi 5099 . . . . 5  |-  ran  ( G  |`  ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
74, 6eqtri 2458 . . . 4  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
87ineq1i 3540 . . 3  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ( ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  i^i 
B )
9 cnvin 5282 . . . . . 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 5010 . . . . . . . . . 10  |-  ( ( A  X.  _V )  i^i  ( _V  X.  B
) )  =  ( ( A  i^i  _V )  X.  ( _V  i^i  B ) )
11 inv1 3656 . . . . . . . . . . 11  |-  ( A  i^i  _V )  =  A
12 incom 3535 . . . . . . . . . . . 12  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
13 inv1 3656 . . . . . . . . . . . 12  |-  ( B  i^i  _V )  =  B
1412, 13eqtri 2458 . . . . . . . . . . 11  |-  ( _V 
i^i  B )  =  B
1511, 14xpeq12i 4903 . . . . . . . . . 10  |-  ( ( A  i^i  _V )  X.  ( _V  i^i  B
) )  =  ( A  X.  B )
1610, 15eqtr2i 2459 . . . . . . . . 9  |-  ( A  X.  B )  =  ( ( A  X.  _V )  i^i  ( _V  X.  B ) )
1716ineq2i 3541 . . . . . . . 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 3555 . . . . . . . 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 5012 . . . . . . . . . . . 12  |-  ( ( Y  i^i  A )  X.  _V )  =  ( ( Y  X.  _V )  i^i  ( A  X.  _V ) )
2019ineq2i 3541 . . . . . . . . . . 11  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( G  i^i  (
( Y  X.  _V )  i^i  ( A  X.  _V ) ) )
21 inass 3553 . . . . . . . . . . 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 2461 . . . . . . . . . 10  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( ( G  i^i  ( Y  X.  _V )
)  i^i  ( A  X.  _V ) )
2322ineq1i 3540 . . . . . . . . 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 3553 . . . . . . . . 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 2458 . . . . . . . 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 2468 . . . . . . 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 5050 . . . . . 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 4893 . . . . . . 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 5293 . . . . . . . 8  |-  `' ( _V  X.  B )  =  ( B  X.  _V )
3029ineq2i 3541 . . . . . . 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 2461 . . . . . 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 2469 . . . . 5  |-  ( `' ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)  |`  B )  =  `' ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
3332dmeqi 5074 . . . 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 3535 . . . . 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 5170 . . . . 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 4892 . . . . . 6  |-  ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  dom  `' ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )
3736ineq1i 3540 . . . . 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 2469 . . . 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 4892 . . . 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 2469 . . 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 2461 . 2  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ran  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
422, 3, 413eqtr4i 2468 1  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ( ( G "
( Y  i^i  A
) )  i^i  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    i^i cin 3321    X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882    |` cres 4883   "cima 4884
This theorem is referenced by:  ecinxp  6982  marypha1lem  7441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894
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