Theorem xpexr2m 4762

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 4745	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4555	. . . . . 6 |- ( E. b b e. B -> dom ( A X. B ) = A )
3	2	adantl 262	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> dom ( A X. B ) = A )
4		dmexg 4596	. . . . . 6 |- ( ( A X. B ) e. C -> dom ( A X. B ) e. _V )
5	4	adantr 261	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> dom ( A X. B ) e. _V )
6	3, 5	eqeltrrd 2115	. . . 4 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> A e. _V )
7		rnxpm 4752	. . . . . 6 |- ( E. a a e. A -> ran ( A X. B ) = B )
8	7	adantl 262	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> ran ( A X. B ) = B )
9		rnexg 4597	. . . . . 6 |- ( ( A X. B ) e. C -> ran ( A X. B ) e. _V )
10	9	adantr 261	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> ran ( A X. B ) e. _V )
11	8, 10	eqeltrrd 2115	. . . 4 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> B e. _V )
12	6, 11	anim12dan 532	. . 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 500	. 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 272	1 |- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   X. cxp 4343   dom cdm 4345   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by: (None)