Theorem xpexr2m 5111

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 5091	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4886	. . . . . 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 4930	. . . . . 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 2274	. . . 4 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> A e. _V )
7		rnxpm 5099	. . . . . 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 4931	. . . . . 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 2274	. . . 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 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   X. cxp 4661   dom cdm 4663   ran crn 4664

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by: (None)