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Theorem dminxp 5276
Description: Domain of the intersection with a Cartesian 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 5026 . . . 4  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  `' ( C  i^i  ( A  X.  B ) )
2 cnvin 5242 . . . . . 6  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  `' ( A  X.  B ) )
3 cnvxp 5253 . . . . . . 7  |-  `' ( A  X.  B )  =  ( B  X.  A )
43ineq2i 3630 . . . . . 6  |-  ( `' C  i^i  `' ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A ) )
52, 4eqtri 2472 . . . . 5  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A
) )
65rneqi 5060 . . . 4  |-  ran  `' ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A ) )
71, 6eqtri 2472 . . 3  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A
) )
87eqeq1i 2455 . 2  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  ran  ( `' C  i^i  ( B  X.  A ) )  =  A )
9 rninxp 5275 . 2  |-  ( ran  ( `' C  i^i  ( B  X.  A
) )  =  A  <->  A. x  e.  A  E. y  e.  B  y `' C x )
10 vex 3047 . . . . 5  |-  y  e. 
_V
11 vex 3047 . . . . 5  |-  x  e. 
_V
1210, 11brcnv 5016 . . . 4  |-  ( y `' C x  <->  x C
y )
1312rexbii 2888 . . 3  |-  ( E. y  e.  B  y `' C x  <->  E. y  e.  B  x C
y )
1413ralbii 2818 . 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 275 1  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  A. x  e.  A  E. y  e.  B  x C
y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1443   A.wral 2736   E.wrex 2737    i^i cin 3402   class class class wbr 4401    X. cxp 4831   `'ccnv 4832   dom cdm 4833   ran crn 4834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-xp 4839  df-rel 4840  df-cnv 4841  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846
This theorem is referenced by:  trust  21237
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