Theorem unixpm 5146

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 4720	. . 3 |- Rel ( A X. B )
2		relfld 5139	. . 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 264	. . . 4 |- ( ( E. b b e. B /\ E. a a e. A ) <-> ( E. a a e. A /\ E. b b e. B ) )
5		xpm 5032	. . . 4 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
6	4, 5	bitri 183	. . 3 |- ( ( E. b b e. B /\ E. a a e. A ) <-> E. x x e. ( A X. B ) )
7		dmxpm 4831	. . . 4 |- ( E. b b e. B -> dom ( A X. B ) = A )
8		rnxpm 5040	. . . 4 |- ( E. a a e. A -> ran ( A X. B ) = B )
9		uneq12 3276	. . . 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 287	. . 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 134	. 2 |- ( E. x x e. ( A X. B ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) = ( A u. B ) )
12	3, 11	eqtrid 2215	1 |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   u. cun 3119   U.cuni 3796   X. cxp 4609   dom cdm 4611   ran crn 4612   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by: (None)