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Theorem dminxp 5283
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 5037 . . . 4  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  `' ( C  i^i  ( A  X.  B ) )
2 cnvin 5249 . . . . . 6  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  `' ( A  X.  B ) )
3 cnvxp 5260 . . . . . . 7  |-  `' ( A  X.  B )  =  ( B  X.  A )
43ineq2i 3554 . . . . . 6  |-  ( `' C  i^i  `' ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A ) )
52, 4eqtri 2463 . . . . 5  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A
) )
65rneqi 5071 . . . 4  |-  ran  `' ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A ) )
71, 6eqtri 2463 . . 3  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A
) )
87eqeq1i 2450 . 2  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  ran  ( `' C  i^i  ( B  X.  A ) )  =  A )
9 rninxp 5282 . 2  |-  ( ran  ( `' C  i^i  ( B  X.  A
) )  =  A  <->  A. x  e.  A  E. y  e.  B  y `' C x )
10 vex 2980 . . . . 5  |-  y  e. 
_V
11 vex 2980 . . . . 5  |-  x  e. 
_V
1210, 11brcnv 5027 . . . 4  |-  ( y `' C x  <->  x C
y )
1312rexbii 2745 . . 3  |-  ( E. y  e.  B  y `' C x  <->  E. y  e.  B  x C
y )
1413ralbii 2744 . 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 271 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 184    = wceq 1369   A.wral 2720   E.wrex 2721    i^i cin 3332   class class class wbr 4297    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
This theorem is referenced by:  trust  19809
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