Theorem xpexr2m 5045

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 5025	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4824	. . . . . 6 |- ( E. b b e. B -> dom ( A X. B ) = A )
3	2	adantl 275	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> dom ( A X. B ) = A )
4		dmexg 4868	. . . . . 6 |- ( ( A X. B ) e. C -> dom ( A X. B ) e. _V )
5	4	adantr 274	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> dom ( A X. B ) e. _V )
6	3, 5	eqeltrrd 2244	. . . 4 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> A e. _V )
7		rnxpm 5033	. . . . . 6 |- ( E. a a e. A -> ran ( A X. B ) = B )
8	7	adantl 275	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> ran ( A X. B ) = B )
9		rnexg 4869	. . . . . 6 |- ( ( A X. B ) e. C -> ran ( A X. B ) e. _V )
10	9	adantr 274	. . . . 5 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> ran ( A X. B ) e. _V )
11	8, 10	eqeltrrd 2244	. . . 4 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> B e. _V )
12	6, 11	anim12dan 590	. . 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 556	. 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 286	1 |- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   X. cxp 4602   dom cdm 4604   ran crn 4605

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)