Theorem ssxpr 3471

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 3462	. . . . . 6 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2		dmxp 3328	. . . . . . 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 3306	. . . . 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 2093	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> A (_ dom ( C X. D))
9		dmxpss 3469	. . . 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 2071	. 2 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> A (_ C)
12		rnxp 3468	. . . . . . 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 3338	. . . . 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 2093	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> B (_ ran ( C X. D))
19		rnxpss 3470	. . . 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 2071	. 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 955   =/= wne 1583   (_ wss 2044  (/)c0 2277   X. cxp 3164  dom cdm 3166  ran crn 3167

This theorem is referenced by:  xp11 3472

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-xp 3180  df-rel 3181  df-cnv 3182  df-dm 3184  df-rn 3185

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