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Theorem unixp 5530
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 5100 . . 3  |-  Rel  ( A  X.  B )
2 relfld 5523 . . 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 5004 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
5 xp0 5415 . . . . 5  |-  ( A  X.  (/) )  =  (/)
64, 5syl6eq 2500 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
76necon3i 2683 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
8 xpeq1 5003 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
9 0xp 5070 . . . . 5  |-  ( (/)  X.  B )  =  (/)
108, 9syl6eq 2500 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1110necon3i 2683 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
12 dmxp 5211 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
13 rnxp 5427 . . . 4  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
14 uneq12 3638 . . . 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 2496 1  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    =/= wne 2638    u. cun 3459   (/)c0 3770   U.cuni 4234    X. cxp 4987   dom cdm 4989   ran crn 4990   Rel wrel 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000
This theorem is referenced by:  unixpid  5532  rankxpl  8296  rankxplim2  8301  rankxplim3  8302  rankxpsuc  8303
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