Theorem ssxpr 3462

Description: A cross-product subclass relationship implies the relationship for it components.

Assertion

Ref	Expression
ssxpr	|- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> (A (_ C /\ B (_ D))

Proof of Theorem ssxpr

Step	Hyp	Ref	Expression
1		xpnz 3452	. . . . . 6 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2		dmxp 3321	. . . . . . 7 |- (B =/= (/) -> dom ( A X. B) = A)
3	2	adantl 388	. . . . . 6 |- ((A =/= (/) /\ B =/= (/)) -> dom ( A X. B) = A)
4	1, 3	sylbir 201	. . . . 5 |- ((A X. B) =/= (/) -> dom ( A X. B) = A)
5	4	adantr 389	. . . 4 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> dom ( A X. B) = A)
6		dmss 3299	. . . . 5 |- ((A X. B) (_ (C X. D) -> dom ( A X. B) (_ dom ( C X. D))
7	6	adantl 388	. . . 4 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> dom ( A X. B) (_ dom ( C X. D))
8	5, 7	eqsstr3d 2086	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> A (_ dom ( C X. D))
9		dmxpss 3459	. . . 4 |- dom ( C X. D) (_ C
10	9	a1i 8	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> dom ( C X. D) (_ C)
11	8, 10	sstrd 2064	. 2 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> A (_ C)
12		rnxp 3458	. . . . . . 7 |- (A =/= (/) -> ran ( A X. B) = B)
13	12	adantr 389	. . . . . 6 |- ((A =/= (/) /\ B =/= (/)) -> ran ( A X. B) = B)
14	1, 13	sylbir 201	. . . . 5 |- ((A X. B) =/= (/) -> ran ( A X. B) = B)
15	14	adantr 389	. . . 4 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> ran ( A X. B) = B)
16		rnss 3331	. . . . 5 |- ((A X. B) (_ (C X. D) -> ran ( A X. B) (_ ran ( C X. D))
17	16	adantl 388	. . . 4 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> ran ( A X. B) (_ ran ( C X. D))
18	15, 17	eqsstr3d 2086	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> B (_ ran ( C X. D))
19		rnxpss 3460	. . . 4 |- ran ( C X. D) (_ D
20	19	a1i 8	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> ran ( C X. D) (_ D)
21	18, 20	sstrd 2064	. 2 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> B (_ D)
22	11, 21	jca 288	1 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> (A (_ C /\ B (_ D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   =/= wne 1577   (_ wss 2037  (/)c0 2270   X. cxp 3158  dom cdm 3160  ran crn 3161

This theorem is referenced by:  xp11 3463

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179

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