ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dminxp Unicode version

Theorem dminxp 4875
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 4628 . . . 4  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  `' ( C  i^i  ( A  X.  B ) )
2 cnvin 4839 . . . . . 6  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  `' ( A  X.  B ) )
3 cnvxp 4850 . . . . . . 7  |-  `' ( A  X.  B )  =  ( B  X.  A )
43ineq2i 3198 . . . . . 6  |-  ( `' C  i^i  `' ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A ) )
52, 4eqtri 2108 . . . . 5  |-  `' ( C  i^i  ( A  X.  B ) )  =  ( `' C  i^i  ( B  X.  A
) )
65rneqi 4663 . . . 4  |-  ran  `' ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A ) )
71, 6eqtri 2108 . . 3  |-  dom  ( C  i^i  ( A  X.  B ) )  =  ran  ( `' C  i^i  ( B  X.  A
) )
87eqeq1i 2095 . 2  |-  ( dom  ( C  i^i  ( A  X.  B ) )  =  A  <->  ran  ( `' C  i^i  ( B  X.  A ) )  =  A )
9 rninxp 4874 . 2  |-  ( ran  ( `' C  i^i  ( B  X.  A
) )  =  A  <->  A. x  e.  A  E. y  e.  B  y `' C x )
10 vex 2622 . . . . 5  |-  y  e. 
_V
11 vex 2622 . . . . 5  |-  x  e. 
_V
1210, 11brcnv 4619 . . . 4  |-  ( y `' C x  <->  x C
y )
1312rexbii 2385 . . 3  |-  ( E. y  e.  B  y `' C x  <->  E. y  e.  B  x C
y )
1413ralbii 2384 . 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 204 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 103    = wceq 1289   A.wral 2359   E.wrex 2360    i^i cin 2998   class class class wbr 3845    X. cxp 4436   `'ccnv 4437   dom cdm 4438   ran crn 4439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator