Theorem ssxpbm 5039

Description: A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
ssxpbm	|- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <-> ( A C_ C /\ B C_ D ) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   C( x)   D( x)

Proof of Theorem ssxpbm

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

Step	Hyp	Ref	Expression
1		xpm 5025	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4824	. . . . . . . . 9 |- ( E. b b e. B -> dom ( A X. B ) = A )
3	2	adantl 275	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) -> dom ( A X. B ) = A )
4	1, 3	sylbir 134	. . . . . . 7 |- ( E. x x e. ( A X. B ) -> dom ( A X. B ) = A )
5	4	adantr 274	. . . . . 6 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> dom ( A X. B ) = A )
6		dmss 4803	. . . . . . 7 |- ( ( A X. B ) C_ ( C X. D ) -> dom ( A X. B ) C_ dom ( C X. D ) )
7	6	adantl 275	. . . . . 6 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> dom ( A X. B ) C_ dom ( C X. D ) )
8	5, 7	eqsstrrd 3179	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ dom ( C X. D ) )
9		dmxpss 5034	. . . . 5 |- dom ( C X. D ) C_ C
10	8, 9	sstrdi 3154	. . . 4 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ C )
11		rnxpm 5033	. . . . . . . . 9 |- ( E. a a e. A -> ran ( A X. B ) = B )
12	11	adantr 274	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) -> ran ( A X. B ) = B )
13	1, 12	sylbir 134	. . . . . . 7 |- ( E. x x e. ( A X. B ) -> ran ( A X. B ) = B )
14	13	adantr 274	. . . . . 6 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> ran ( A X. B ) = B )
15		rnss 4834	. . . . . . 7 |- ( ( A X. B ) C_ ( C X. D ) -> ran ( A X. B ) C_ ran ( C X. D ) )
16	15	adantl 275	. . . . . 6 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> ran ( A X. B ) C_ ran ( C X. D ) )
17	14, 16	eqsstrrd 3179	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ ran ( C X. D ) )
18		rnxpss 5035	. . . . 5 |- ran ( C X. D ) C_ D
19	17, 18	sstrdi 3154	. . . 4 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ D )
20	10, 19	jca 304	. . 3 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> ( A C_ C /\ B C_ D ) )
21	20	ex 114	. 2 |- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) -> ( A C_ C /\ B C_ D ) ) )
22		xpss12 4711	. 2 |- ( ( A C_ C /\ B C_ D ) -> ( A X. B ) C_ ( C X. D ) )
23	21, 22	impbid1 141	1 |- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <-> ( A C_ C /\ B C_ D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

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-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:  xp11m  5042