MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imainrect Unicode version

Theorem imainrect 5279
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 4857 . . 3  |-  ( ( G  i^i  ( A  X.  B ) )  |`  Y )  =  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
21rneqi 5063 . 2  |-  ran  (
( G  i^i  ( A  X.  B ) )  |`  Y )  =  ran  ( ( G  i^i  ( A  X.  B
) )  i^i  ( Y  X.  _V ) )
3 df-ima 4858 . 2  |-  ( ( G  i^i  ( A  X.  B ) )
" Y )  =  ran  ( ( G  i^i  ( A  X.  B ) )  |`  Y )
4 df-ima 4858 . . . . 5  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  |`  ( Y  i^i  A ) )
5 df-res 4857 . . . . . 6  |-  ( G  |`  ( Y  i^i  A
) )  =  ( G  i^i  ( ( Y  i^i  A )  X.  _V ) )
65rneqi 5063 . . . . 5  |-  ran  ( G  |`  ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
74, 6eqtri 2432 . . . 4  |-  ( G
" ( Y  i^i  A ) )  =  ran  ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)
87ineq1i 3506 . . 3  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ( ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  i^i 
B )
9 cnvin 5246 . . . . . 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 4974 . . . . . . . . . 10  |-  ( ( A  X.  _V )  i^i  ( _V  X.  B
) )  =  ( ( A  i^i  _V )  X.  ( _V  i^i  B ) )
11 inv1 3622 . . . . . . . . . . 11  |-  ( A  i^i  _V )  =  A
12 incom 3501 . . . . . . . . . . . 12  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
13 inv1 3622 . . . . . . . . . . . 12  |-  ( B  i^i  _V )  =  B
1412, 13eqtri 2432 . . . . . . . . . . 11  |-  ( _V 
i^i  B )  =  B
1511, 14xpeq12i 4867 . . . . . . . . . 10  |-  ( ( A  i^i  _V )  X.  ( _V  i^i  B
) )  =  ( A  X.  B )
1610, 15eqtr2i 2433 . . . . . . . . 9  |-  ( A  X.  B )  =  ( ( A  X.  _V )  i^i  ( _V  X.  B ) )
1716ineq2i 3507 . . . . . . . 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 3521 . . . . . . . 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 4976 . . . . . . . . . . . 12  |-  ( ( Y  i^i  A )  X.  _V )  =  ( ( Y  X.  _V )  i^i  ( A  X.  _V ) )
2019ineq2i 3507 . . . . . . . . . . 11  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( G  i^i  (
( Y  X.  _V )  i^i  ( A  X.  _V ) ) )
21 inass 3519 . . . . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  ( ( G  i^i  ( Y  X.  _V )
)  i^i  ( A  X.  _V ) )
2322ineq1i 3506 . . . . . . . . 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 3519 . . . . . . . . 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 2432 . . . . . . . 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 2442 . . . . . . 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 5014 . . . . . 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 4857 . . . . . . 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 5257 . . . . . . . 8  |-  `' ( _V  X.  B )  =  ( B  X.  _V )
3029ineq2i 3507 . . . . . . 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 2435 . . . . . 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 2443 . . . . 5  |-  ( `' ( G  i^i  (
( Y  i^i  A
)  X.  _V )
)  |`  B )  =  `' ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
3332dmeqi 5038 . . . 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 3501 . . . . 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 5134 . . . . 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 4856 . . . . . 6  |-  ran  ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )  =  dom  `' ( G  i^i  ( ( Y  i^i  A )  X. 
_V ) )
3736ineq1i 3506 . . . . 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 2443 . . . 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 4856 . . . 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 2443 . . 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 2435 . 2  |-  ( ( G " ( Y  i^i  A ) )  i^i  B )  =  ran  ( ( G  i^i  ( A  X.  B ) )  i^i  ( Y  X.  _V ) )
422, 3, 413eqtr4i 2442 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 1649   _Vcvv 2924    i^i cin 3287    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848
This theorem is referenced by:  ecinxp  6946  marypha1lem  7404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
  Copyright terms: Public domain W3C validator