Theorem xpexr2m 5123

Description: If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)

Assertion

Ref	Expression
xpexr2m	|- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) -> ( A e. _V /\ B e. _V ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem xpexr2m

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

Step	Hyp	Ref	Expression
1		xpm 5103	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4897	. . . . . 6 |- ( E. b b e. B -> dom ( A X. B ) = A )
3	2	adantl 277	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> dom ( A X. B ) = A )
4		dmexg 4941	. . . . . 6 |- ( ( A X. B ) e. C -> dom ( A X. B ) e. _V )
5	4	adantr 276	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> dom ( A X. B ) e. _V )
6	3, 5	eqeltrrd 2282	. . . 4 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> A e. _V )
7		rnxpm 5111	. . . . . 6 |- ( E. a a e. A -> ran ( A X. B ) = B )
8	7	adantl 277	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> ran ( A X. B ) = B )
9		rnexg 4942	. . . . . 6 |- ( ( A X. B ) e. C -> ran ( A X. B ) e. _V )
10	9	adantr 276	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> ran ( A X. B ) e. _V )
11	8, 10	eqeltrrd 2282	. . . 4 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> B e. _V )
12	6, 11	anim12dan 600	. . 3 |- ( ( ( A X. B ) e. C /\ ( E. b b e. B /\ E. a a e. A ) ) -> ( A e. _V /\ B e. _V ) )
13	12	ancom2s 566	. 2 |- ( ( ( A X. B ) e. C /\ ( E. a a e. A /\ E. b b e. B ) ) -> ( A e. _V /\ B e. _V ) )
14	1, 13	sylan2br 288	1 |- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   E.wex 1514   e. wcel 2175   _Vcvv 2771   X. cxp 4672   dom cdm 4674   ran crn 4675

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-dm 4684  df-rn 4685

This theorem is referenced by: (None)