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Theorem dminxp 5118
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 4872 . . . 4  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  `' ( C  i^i  ( A  X.  B ) )
2 cnvin 5088 . . . . . 6  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  `' ( A  X.  B ) )
3 cnvxp 5097 . . . . . . 7  |-  `' ( A  X.  B )  =  ( B  X.  A )
43ineq2i 3367 . . . . . 6  |-  ( `' C  i^i  `' ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A ) )
52, 4eqtri 2303 . . . . 5  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A
) )
65rneqi 4905 . . . 4  |-  ran  `' ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A ) )
71, 6eqtri 2303 . . 3  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A
) )
87eqeq1i 2290 . 2  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  ran  ( `' C  i^i  ( B  X.  A ) )  =  A )
9 rninxp 5117 . 2  |-  ( ran  ( `' C  i^i  ( B  X.  A
) )  =  A  <->  A. x  e.  A  E. y  e.  B  y `' C x )
10 vex 2791 . . . . 5  |-  y  e. 
_V
11 vex 2791 . . . . 5  |-  x  e. 
_V
1210, 11brcnv 4864 . . . 4  |-  ( y `' C x  <->  x C
y )
1312rexbii 2568 . . 3  |-  ( E. y  e.  B  y `' C x  <->  E. y  e.  B  x C
y )
1413ralbii 2567 . 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 262 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 176    = wceq 1623   A.wral 2543   E.wrex 2544    i^i cin 3151   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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