Theorem unixpm 5166

Description: The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
unixpm	|- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem unixpm

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 4737	. . 3 |- Rel ( A X. B )
2		relfld 5159	. . 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		ancom 266	. . . 4 |- ( ( E. b b e. B /\ E. a a e. A ) <-> ( E. a a e. A /\ E. b b e. B ) )
5		xpm 5052	. . . 4 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
6	4, 5	bitri 184	. . 3 |- ( ( E. b b e. B /\ E. a a e. A ) <-> E. x x e. ( A X. B ) )
7		dmxpm 4849	. . . 4 |- ( E. b b e. B -> dom ( A X. B ) = A )
8		rnxpm 5060	. . . 4 |- ( E. a a e. A -> ran ( A X. B ) = B )
9		uneq12 3286	. . . 4 |- ( ( dom ( A X. B ) = A /\ ran ( A X. B ) = B ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
10	7, 8, 9	syl2an 289	. . 3 |- ( ( E. b b e. B /\ E. a a e. A ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
11	6, 10	sylbir 135	. 2 |- ( E. x x e. ( A X. B ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
12	3, 11	eqtrid 2222	1 |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   u. cun 3129   U.cuni 3811   X. cxp 4626   dom cdm 4628   ran crn 4629   Rel wrel 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by: (None)