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Theorem imainrect 5222
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 4804 . . 3  |-  ( ( G  i^i  ( A  X.  B ) )  |`  Y )  =  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
21rneqi 5008 . 2  |-  ran  (
( G  i^i  ( A  X.  B ) )  |`  Y )  =  ran  ( ( G  i^i  ( A  X.  B
) )  i^i  ( Y  X.  _V ) )
3 df-ima 4805 . 2  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ran  ( ( G  i^i  ( A  X.  B ) )  |`  Y )
4 df-ima 4805 . . . . 5  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  |`  ( Y  i^i  A ) )
5 df-res 4804 . . . . . 6  |-  ( G  |`  ( Y  i^i  A
) )  =  ( G  i^i  ( ( Y  i^i  A )  X.  _V ) )
65rneqi 5008 . . . . 5  |-  ran  ( G  |`  ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
74, 6eqtri 2386 . . . 4  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
87ineq1i 3454 . . 3  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ( ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  i^i 
B )
9 cnvin 5191 . . . . . 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 4921 . . . . . . . . . 10  |-  ( ( A  X.  _V )  i^i  ( _V  X.  B
) )  =  ( ( A  i^i  _V )  X.  ( _V  i^i  B ) )
11 inv1 3569 . . . . . . . . . . 11  |-  ( A  i^i  _V )  =  A
12 incom 3449 . . . . . . . . . . . 12  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
13 inv1 3569 . . . . . . . . . . . 12  |-  ( B  i^i  _V )  =  B
1412, 13eqtri 2386 . . . . . . . . . . 11  |-  ( _V 
i^i  B )  =  B
1511, 14xpeq12i 4814 . . . . . . . . . 10  |-  ( ( A  i^i  _V )  X.  ( _V  i^i  B
) )  =  ( A  X.  B )
1610, 15eqtr2i 2387 . . . . . . . . 9  |-  ( A  X.  B )  =  ( ( A  X.  _V )  i^i  ( _V  X.  B ) )
1716ineq2i 3455 . . . . . . . 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 3469 . . . . . . . 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 4923 . . . . . . . . . . . 12  |-  ( ( Y  i^i  A )  X.  _V )  =  ( ( Y  X.  _V )  i^i  ( A  X.  _V ) )
2019ineq2i 3455 . . . . . . . . . . 11  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( G  i^i  (
( Y  X.  _V )  i^i  ( A  X.  _V ) ) )
21 inass 3467 . . . . . . . . . . 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 2389 . . . . . . . . . 10  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( ( G  i^i  ( Y  X.  _V )
)  i^i  ( A  X.  _V ) )
2322ineq1i 3454 . . . . . . . . 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 3467 . . . . . . . . 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 2386 . . . . . . . 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 2396 . . . . . . 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 4959 . . . . . 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 4804 . . . . . . 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 5200 . . . . . . . 8  |-  `' ( _V  X.  B )  =  ( B  X.  _V )
3029ineq2i 3455 . . . . . . 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 2389 . . . . . 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 2397 . . . . 5  |-  ( `' ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)  |`  B )  =  `' ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
3332dmeqi 4983 . . . 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 3449 . . . . 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 4803 . . . . . 6  |-  ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  dom  `' ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )
3736ineq1i 3454 . . . . 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 2397 . . . 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 4803 . . . 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 2397 . . 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 2389 . 2  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ran  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
422, 3, 413eqtr4i 2396 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 1647   _Vcvv 2873    i^i cin 3237    X. cxp 4790   `'ccnv 4791   dom cdm 4792   ran crn 4793    |` cres 4794   "cima 4795
This theorem is referenced by:  ecinxp  6876  marypha1lem  7333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805
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