Theorem unixpm 5272

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 4835	. . 3 |- Rel ( A X. B )
2		relfld 5265	. . 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 5158	. . . 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 4952	. . . 4 |- ( E. b b e. B -> dom ( A X. B ) = A )
8		rnxpm 5166	. . . 4 |- ( E. a a e. A -> ran ( A X. B ) = B )
9		uneq12 3356	. . . 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 2276	1 |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   u. cun 3198   U.cuni 3893   X. cxp 4723   dom cdm 4725   ran crn 4726   Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)