Theorem ssxpbm 5118

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 5104	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 4898	. . . . . . . . 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 4877	. . . . . . 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 3230	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ dom ( C X. D ) )
9		dmxpss 5113	. . . . 5 |- dom ( C X. D ) C_ C
10	8, 9	sstrdi 3205	. . . 4 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ C )
11		rnxpm 5112	. . . . . . . . 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 4908	. . . . . . 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 3230	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ ran ( C X. D ) )
18		rnxpss 5114	. . . . 5 |- ran ( C X. D ) C_ D
19	17, 18	sstrdi 3205	. . . 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 4782	. 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 1373   E.wex 1515   e. wcel 2176   C_ wss 3166   X. cxp 4673   dom cdm 4675   ran crn 4676

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686

This theorem is referenced by:  xp11m  5121