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Theorem dminxp 5253
Description: Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
dminxp  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  A. x  e.  A  E. y  e.  B  x C
y )
Distinct variable groups:    x, A    x, y, B    x, C, y
Allowed substitution hint:    A( y)

Proof of Theorem dminxp
StepHypRef Expression
1 dfdm4 5005 . . . 4  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  `' ( C  i^i  ( A  X.  B ) )
2 cnvin 5221 . . . . . 6  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  `' ( A  X.  B ) )
3 cnvxp 5232 . . . . . . 7  |-  `' ( A  X.  B )  =  ( B  X.  A )
43ineq2i 3484 . . . . . 6  |-  ( `' C  i^i  `' ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A ) )
52, 4eqtri 2409 . . . . 5  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A
) )
65rneqi 5038 . . . 4  |-  ran  `' ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A ) )
71, 6eqtri 2409 . . 3  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A
) )
87eqeq1i 2396 . 2  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  ran  ( `' C  i^i  ( B  X.  A ) )  =  A )
9 rninxp 5252 . 2  |-  ( ran  ( `' C  i^i  ( B  X.  A
) )  =  A  <->  A. x  e.  A  E. y  e.  B  y `' C x )
10 vex 2904 . . . . 5  |-  y  e. 
_V
11 vex 2904 . . . . 5  |-  x  e. 
_V
1210, 11brcnv 4997 . . . 4  |-  ( y `' C x  <->  x C
y )
1312rexbii 2676 . . 3  |-  ( E. y  e.  B  y `' C x  <->  E. y  e.  B  x C
y )
1413ralbii 2675 . 2  |-  ( A. x  e.  A  E. y  e.  B  y `' C x  <->  A. x  e.  A  E. y  e.  B  x C
y )
158, 9, 143bitri 263 1  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  A. x  e.  A  E. y  e.  B  x C
y )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   A.wral 2651   E.wrex 2652    i^i cin 3264   class class class wbr 4155    X. cxp 4818   `'ccnv 4819   dom cdm 4820   ran crn 4821
This theorem is referenced by:  trust  18182
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833
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