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Theorem unixp 5404
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 4985 . . 3  |-  Rel  ( A  X.  B )
2 relfld 5397 . . 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 4895 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
5 xp0 5293 . . . . 5  |-  ( A  X.  (/) )  =  (/)
64, 5syl6eq 2486 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
76necon3i 2645 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
8 xpeq1 4894 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
9 xp0r 4958 . . . . 5  |-  ( (/)  X.  B )  =  (/)
108, 9syl6eq 2486 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1110necon3i 2645 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
12 dmxp 5090 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
13 rnxp 5301 . . . 4  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
14 uneq12 3498 . . . 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 465 . . 3  |-  ( ( B  =/=  (/)  /\  A  =/=  (/) )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
167, 11, 15syl2anc 644 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
173, 16syl5eq 2482 1  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    =/= wne 2601    u. cun 3320   (/)c0 3630   U.cuni 4017    X. cxp 4878   dom cdm 4880   ran crn 4881   Rel wrel 4885
This theorem is referenced by:  unixpid  5406  rankxpl  7803  rankxplim2  7806  rankxplim3  7807  rankxpsuc  7808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891
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