Theorem unixp 5668

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

Step	Hyp	Ref	Expression
1		relxp 5227	. . 3 |- Rel ( A X. B )
2		relfld 5661	. . 3 |- ( Rel ( A X. B ) -> U. U. ( A X. B ) = ( dom ( A X. B ) u. ran ( A X. B ) ) )
3	1, 2	ax-mp 5	. 2 |- U. U. ( A X. B ) = ( dom ( A X. B ) u. ran ( A X. B ) )
4		xpeq2 5129	. . . . 5 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
5		xp0 5552	. . . . 5 |- ( A X. (/) ) = (/)
6	4, 5	syl6eq 2672	. . . 4 |- ( B = (/) -> ( A X. B ) = (/) )
7	6	necon3i 2826	. . 3 |- ( ( A X. B ) =/= (/) -> B =/= (/) )
8		xpeq1 5128	. . . . 5 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
9		0xp 5199	. . . . 5 |- ( (/) X. B ) = (/)
10	8, 9	syl6eq 2672	. . . 4 |- ( A = (/) -> ( A X. B ) = (/) )
11	10	necon3i 2826	. . 3 |- ( ( A X. B ) =/= (/) -> A =/= (/) )
12		dmxp 5344	. . . 4 |- ( B =/= (/) -> dom ( A X. B ) = A )
13		rnxp 5564	. . . 4 |- ( A =/= (/) -> ran ( A X. B ) = B )
14		uneq12 3762	. . . 4 |- ( ( dom ( A X. B ) = A /\ ran ( A X. B ) = B ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
15	12, 13, 14	syl2an 494	. . 3 |- ( ( B =/= (/) /\ A =/= (/) ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
16	7, 11, 15	syl2anc 693	. 2 |- ( ( A X. B ) =/= (/) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
17	3, 16	syl5eq 2668	1 |- ( ( A X. B ) =/= (/) -> U. U. ( A X. B ) = ( A u. B ) )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   u. cun 3572   (/)c0 3915   U.cuni 4436   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  unixpid  5670  rankxpl  8738  rankxplim2  8743  rankxplim3  8744  rankxpsuc  8745