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Theorem unixpm 5044
Description: The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
unixpm  |-  ( E. x  x  e.  ( A  X.  B )  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem unixpm
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4618 . . 3  |-  Rel  ( A  X.  B )
2 relfld 5037 . . 3  |-  ( Rel  ( A  X.  B
)  ->  U. U. ( A  X.  B )  =  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) )
31, 2ax-mp 5 . 2  |-  U. U. ( A  X.  B
)  =  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )
4 ancom 264 . . . 4  |-  ( ( E. b  b  e.  B  /\  E. a 
a  e.  A )  <-> 
( E. a  a  e.  A  /\  E. b  b  e.  B
) )
5 xpm 4930 . . . 4  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. x  x  e.  ( A  X.  B
) )
64, 5bitri 183 . . 3  |-  ( ( E. b  b  e.  B  /\  E. a 
a  e.  A )  <->  E. x  x  e.  ( A  X.  B
) )
7 dmxpm 4729 . . . 4  |-  ( E. b  b  e.  B  ->  dom  ( A  X.  B )  =  A )
8 rnxpm 4938 . . . 4  |-  ( E. a  a  e.  A  ->  ran  ( A  X.  B )  =  B )
9 uneq12 3195 . . . 4  |-  ( ( dom  ( A  X.  B )  =  A  /\  ran  ( A  X.  B )  =  B )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
107, 8, 9syl2an 287 . . 3  |-  ( ( E. b  b  e.  B  /\  E. a 
a  e.  A )  ->  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) )  =  ( A  u.  B
) )
116, 10sylbir 134 . 2  |-  ( E. x  x  e.  ( A  X.  B )  ->  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) )  =  ( A  u.  B
) )
123, 11syl5eq 2162 1  |-  ( E. x  x  e.  ( A  X.  B )  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465    u. cun 3039   U.cuni 3706    X. cxp 4507   dom cdm 4509   ran crn 4510   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by: (None)
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