Theorem unixpm 5223

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 4788	. . 3 |- Rel ( A X. B )
2		relfld 5216	. . 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 5109	. . . 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 4903	. . . 4 |- ( E. b b e. B -> dom ( A X. B ) = A )
8		rnxpm 5117	. . . 4 |- ( E. a a e. A -> ran ( A X. B ) = B )
9		uneq12 3323	. . . 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 2251	1 |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2177   u. cun 3165   U.cuni 3852   X. cxp 4677   dom cdm 4679   ran crn 4680   Rel wrel 4684

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-xp 4685  df-rel 4686  df-cnv 4687  df-dm 4689  df-rn 4690

This theorem is referenced by: (None)