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Theorem unixp 5533
Description: The double class union of a nonempty Cartesian 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 5103 . . 3  |-  Rel  ( A  X.  B )
2 relfld 5526 . . 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 xpeq2 5009 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
5 xp0 5418 . . . . 5  |-  ( A  X.  (/) )  =  (/)
64, 5syl6eq 2519 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
76necon3i 2702 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
8 xpeq1 5008 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
9 0xp 5073 . . . . 5  |-  ( (/)  X.  B )  =  (/)
108, 9syl6eq 2519 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1110necon3i 2702 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
12 dmxp 5214 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
13 rnxp 5430 . . . 4  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
14 uneq12 3648 . . . 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 477 . . 3  |-  ( ( B  =/=  (/)  /\  A  =/=  (/) )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
167, 11, 15syl2anc 661 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
173, 16syl5eq 2515 1  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    =/= wne 2657    u. cun 3469   (/)c0 3780   U.cuni 4240    X. cxp 4992   dom cdm 4994   ran crn 4995   Rel wrel 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rn 5005
This theorem is referenced by:  unixpid  5535  rankxpl  8284  rankxplim2  8289  rankxplim3  8290  rankxpsuc  8291
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