Theorem unixpm 5182

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 4753	. . 3 |- Rel ( A X. B )
2		relfld 5175	. . 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 5068	. . . 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 4865	. . . 4 |- ( E. b b e. B -> dom ( A X. B ) = A )
8		rnxpm 5076	. . . 4 |- ( E. a a e. A -> ran ( A X. B ) = B )
9		uneq12 3299	. . . 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 2234	1 |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   u. cun 3142   U.cuni 3824   X. cxp 4642   dom cdm 4644   ran crn 4645   Rel wrel 4649

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655

This theorem is referenced by: (None)