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Theorem imainrect 5438
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 5001 . . 3  |-  ( ( G  i^i  ( A  X.  B ) )  |`  Y )  =  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
21rneqi 5219 . 2  |-  ran  (
( G  i^i  ( A  X.  B ) )  |`  Y )  =  ran  ( ( G  i^i  ( A  X.  B
) )  i^i  ( Y  X.  _V ) )
3 df-ima 5002 . 2  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ran  ( ( G  i^i  ( A  X.  B ) )  |`  Y )
4 df-ima 5002 . . . . 5  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  |`  ( Y  i^i  A ) )
5 df-res 5001 . . . . . 6  |-  ( G  |`  ( Y  i^i  A
) )  =  ( G  i^i  ( ( Y  i^i  A )  X.  _V ) )
65rneqi 5219 . . . . 5  |-  ran  ( G  |`  ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
74, 6eqtri 2472 . . . 4  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
87ineq1i 3681 . . 3  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ( ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  i^i 
B )
9 cnvin 5403 . . . . . 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 5125 . . . . . . . . . 10  |-  ( ( A  X.  _V )  i^i  ( _V  X.  B
) )  =  ( ( A  i^i  _V )  X.  ( _V  i^i  B ) )
11 inv1 3798 . . . . . . . . . . 11  |-  ( A  i^i  _V )  =  A
12 incom 3676 . . . . . . . . . . . 12  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
13 inv1 3798 . . . . . . . . . . . 12  |-  ( B  i^i  _V )  =  B
1412, 13eqtri 2472 . . . . . . . . . . 11  |-  ( _V 
i^i  B )  =  B
1511, 14xpeq12i 5011 . . . . . . . . . 10  |-  ( ( A  i^i  _V )  X.  ( _V  i^i  B
) )  =  ( A  X.  B )
1610, 15eqtr2i 2473 . . . . . . . . 9  |-  ( A  X.  B )  =  ( ( A  X.  _V )  i^i  ( _V  X.  B ) )
1716ineq2i 3682 . . . . . . . 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 3695 . . . . . . . 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 5127 . . . . . . . . . . . 12  |-  ( ( Y  i^i  A )  X.  _V )  =  ( ( Y  X.  _V )  i^i  ( A  X.  _V ) )
2019ineq2i 3682 . . . . . . . . . . 11  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( G  i^i  (
( Y  X.  _V )  i^i  ( A  X.  _V ) ) )
21 inass 3693 . . . . . . . . . . 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 2475 . . . . . . . . . 10  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( ( G  i^i  ( Y  X.  _V )
)  i^i  ( A  X.  _V ) )
2322ineq1i 3681 . . . . . . . . 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 3693 . . . . . . . . 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 2472 . . . . . . . 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 2482 . . . . . . 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 5167 . . . . . 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 5001 . . . . . . 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 5414 . . . . . . . 8  |-  `' ( _V  X.  B )  =  ( B  X.  _V )
3029ineq2i 3682 . . . . . . 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 2475 . . . . . 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 2483 . . . . 5  |-  ( `' ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)  |`  B )  =  `' ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
3332dmeqi 5194 . . . 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 3676 . . . . 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 5284 . . . . 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 5000 . . . . . 6  |-  ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  dom  `' ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )
3736ineq1i 3681 . . . . 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 2483 . . . 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 5000 . . . 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 2483 . . 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 2475 . 2  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ran  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
422, 3, 413eqtr4i 2482 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 1383   _Vcvv 3095    i^i cin 3460    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002
This theorem is referenced by:  ecinxp  7388  marypha1lem  7895
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