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Theorem unixp 5205
Description: The double class union of a non-empty cross product is the union of it members. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
unixp  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )

Proof of Theorem unixp
StepHypRef Expression
1 relxp 4794 . . 3  |-  Rel  ( A  X.  B )
2 relfld 5198 . . 3  |-  ( Rel  ( A  X.  B
)  ->  U. U. ( A  X.  B )  =  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) )
31, 2ax-mp 8 . 2  |-  U. U. ( A  X.  B
)  =  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )
4 xpeq2 4704 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
5 xp0 5098 . . . . 5  |-  ( A  X.  (/) )  =  (/)
64, 5syl6eq 2331 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
76necon3i 2485 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
8 xpeq1 4703 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
9 xp0r 4768 . . . . 5  |-  ( (/)  X.  B )  =  (/)
108, 9syl6eq 2331 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1110necon3i 2485 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
12 dmxp 4897 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
13 rnxp 5106 . . . 4  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
14 uneq12 3324 . . . 4  |-  ( ( dom  ( A  X.  B )  =  A  /\  ran  ( A  X.  B )  =  B )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
1512, 13, 14syl2an 463 . . 3  |-  ( ( B  =/=  (/)  /\  A  =/=  (/) )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
167, 11, 15syl2anc 642 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
173, 16syl5eq 2327 1  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    =/= wne 2446    u. cun 3150   (/)c0 3455   U.cuni 3827    X. cxp 4687   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  unixpid  5207  rankxpl  7547  rankxplim2  7550  rankxplim3  7551  rankxpsuc  7552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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