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Theorem unixpm 4877
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 4469 . . 3  |-  Rel  ( A  X.  B )
2 relfld 4870 . . 3  |-  ( Rel  ( A  X.  B
)  ->  U. U. ( A  X.  B )  =  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) )
31, 2ax-mp 7 . 2  |-  U. U. ( A  X.  B
)  =  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )
4 ancom 262 . . . 4  |-  ( ( E. b  b  e.  B  /\  E. a 
a  e.  A )  <-> 
( E. a  a  e.  A  /\  E. b  b  e.  B
) )
5 xpm 4769 . . . 4  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. x  x  e.  ( A  X.  B
) )
64, 5bitri 182 . . 3  |-  ( ( E. b  b  e.  B  /\  E. a 
a  e.  A )  <->  E. x  x  e.  ( A  X.  B
) )
7 dmxpm 4577 . . . 4  |-  ( E. b  b  e.  B  ->  dom  ( A  X.  B )  =  A )
8 rnxpm 4776 . . . 4  |-  ( E. a  a  e.  A  ->  ran  ( A  X.  B )  =  B )
9 uneq12 3122 . . . 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 283 . . 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 133 . 2  |-  ( E. x  x  e.  ( A  X.  B )  ->  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) )  =  ( A  u.  B
) )
123, 11syl5eq 2126 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 102    = wceq 1285   E.wex 1422    e. wcel 1434    u. cun 2972   U.cuni 3603    X. cxp 4363   dom cdm 4365   ran crn 4366   Rel wrel 4370
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-dm 4375  df-rn 4376
This theorem is referenced by: (None)
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