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Theorem unixp 5380
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 4953 . . 3  |-  Rel  ( A  X.  B )
2 relfld 5372 . . 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 4860 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
5 xp0 5266 . . . . 5  |-  ( A  X.  (/) )  =  (/)
64, 5syl6eq 2477 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
76necon3i 2662 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
8 xpeq1 4859 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
9 0xp 4926 . . . . 5  |-  ( (/)  X.  B )  =  (/)
108, 9syl6eq 2477 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1110necon3i 2662 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
12 dmxp 5064 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
13 rnxp 5278 . . . 4  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
14 uneq12 3612 . . . 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 479 . . 3  |-  ( ( B  =/=  (/)  /\  A  =/=  (/) )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
167, 11, 15syl2anc 665 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  =  ( A  u.  B ) )
173, 16syl5eq 2473 1  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    =/= wne 2616    u. cun 3431   (/)c0 3758   U.cuni 4213    X. cxp 4843   dom cdm 4845   ran crn 4846   Rel wrel 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856
This theorem is referenced by:  unixpid  5382  rankxpl  8336  rankxplim2  8341  rankxplim3  8342  rankxpsuc  8343
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