Theorem ssxpbm 5221

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 5207	. . . . . . . 8 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. x x e. ( A X. B ) )
2		dmxpm 5000	. . . . . . . . 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 4978	. . . . . . 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 3285	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ dom ( C X. D ) )
9		dmxpss 5216	. . . . 5 |- dom ( C X. D ) C_ C
10	8, 9	sstrdi 3260	. . . 4 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ C )
11		rnxpm 5215	. . . . . . . . 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 5010	. . . . . . 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 3285	. . . . 5 |- ( ( E. x x e. ( A X. B ) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ ran ( C X. D ) )
18		rnxpss 5217	. . . . 5 |- ran ( C X. D ) C_ D
19	17, 18	sstrdi 3260	. . . 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 4880	. 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 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   X. cxp 4770   dom cdm 4772   ran crn 4773

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 4247  ax-pow 4309  ax-pr 4344

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-dm 4782  df-rn 4783

This theorem is referenced by:  xp11m  5224