Theorem ssxpbm 5200

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 5186	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4979	. . . . . . . . 9 |- ( E. b b e. B -> dom ( A X. B ) = A )
3	2	adantl 277	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) -> dom ( A X. B ) = A )
4	1, 3	sylbir 135	. . . . . . 7 |- ( E. x x e. ( A X. B ) -> dom ( A X. B ) = A )
5	4	adantr 276	. . . . . 6 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> dom ( A X. B ) = A )
6		dmss 4957	. . . . . . 7 |- ( ( A X. B ) C_ ( C X. D ) -> dom ( A X. B ) C_ dom ( C X. D ) )
7	6	adantl 277	. . . . . 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 3277	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ dom ( C X. D ) )
9		dmxpss 5195	. . . . 5 |- dom ( C X. D ) C_ C
10	8, 9	sstrdi 3252	. . . 4 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ C )
11		rnxpm 5194	. . . . . . . . 9 |- ( E. a a e. A -> ran ( A X. B ) = B )
12	11	adantr 276	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) -> ran ( A X. B ) = B )
13	1, 12	sylbir 135	. . . . . . 7 |- ( E. x x e. ( A X. B ) -> ran ( A X. B ) = B )
14	13	adantr 276	. . . . . 6 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> ran ( A X. B ) = B )
15		rnss 4989	. . . . . . 7 |- ( ( A X. B ) C_ ( C X. D ) -> ran ( A X. B ) C_ ran ( C X. D ) )
16	15	adantl 277	. . . . . 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 3277	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ ran ( C X. D ) )
18		rnxpss 5196	. . . . 5 |- ran ( C X. D ) C_ D
19	17, 18	sstrdi 3252	. . . 4 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ D )
20	10, 19	jca 306	. . 3 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> ( A C_ C /\ B C_ D ) )
21	20	ex 115	. 2 |- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) -> ( A C_ C /\ B C_ D ) ) )
22		xpss12 4859	. 2 |- ( ( A C_ C /\ B C_ D ) -> ( A X. B ) C_ ( C X. D ) )
23	21, 22	impbid1 142	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 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   C_ wss 3213   X. cxp 4749   dom cdm 4751   ran crn 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762

This theorem is referenced by:  xp11m  5203