Theorem xpexr2m 5224

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 5204	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4997	. . . . . 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 5041	. . . . . 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 2316	. . . 4 |- ( ( ( A X. B ) e. C /\ E. b b e. B ) -> A e. _V )
7		rnxpm 5212	. . . . . 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 5042	. . . . . 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 2316	. . . 4 |- ( ( ( A X. B ) e. C /\ E. a a e. A ) -> B e. _V )
12	6, 11	anim12dan 608	. . 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 572	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   X. cxp 4767   dom cdm 4769   ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by: (None)