Theorem ssxpb 4346

Description: A cross-product subclass relationship is equivalent to the relationship for it components.

Assertion

Ref	Expression
ssxpb	|- ((A X. B) =/= (/) -> ((A X. B) C_ (C X. D) <-> (A C_ C /\ B C_ D)))

Proof of Theorem ssxpb

Step	Hyp	Ref	Expression
1		xpnz 4335	. . . . . . . 8 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2		dmxp 4177	. . . . . . . . 9 |- (B =/= (/) -> dom ( A X. B) = A)
3	2	adantl 424	. . . . . . . 8 |- ((A =/= (/) /\ B =/= (/)) -> dom ( A X. B) = A)
4	1, 3	sylbir 218	. . . . . . 7 |- ((A X. B) =/= (/) -> dom ( A X. B) = A)
5	4	adantr 425	. . . . . 6 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> dom ( A X. B) = A)
6		dmss 4156	. . . . . . 7 |- ((A X. B) C_ (C X. D) -> dom ( A X. B) C_ dom ( C X. D))
7	6	adantl 424	. . . . . 6 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> dom ( A X. B) C_ dom ( C X. D))
8	5, 7	eqsstr3d 2652	. . . . 5 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> A C_ dom ( C X. D))
9		dmxpss 4343	. . . . . 6 |- dom ( C X. D) C_ C
10	9	a1i 8	. . . . 5 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> dom ( C X. D) C_ C)
11	8, 10	sstrd 2627	. . . 4 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> A C_ C)
12		rnxp 4342	. . . . . . . . 9 |- (A =/= (/) -> ran ( A X. B) = B)
13	12	adantr 425	. . . . . . . 8 |- ((A =/= (/) /\ B =/= (/)) -> ran ( A X. B) = B)
14	1, 13	sylbir 218	. . . . . . 7 |- ((A X. B) =/= (/) -> ran ( A X. B) = B)
15	14	adantr 425	. . . . . 6 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> ran ( A X. B) = B)
16		rnss 4189	. . . . . . 7 |- ((A X. B) C_ (C X. D) -> ran ( A X. B) C_ ran ( C X. D))
17	16	adantl 424	. . . . . 6 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> ran ( A X. B) C_ ran ( C X. D))
18	15, 17	eqsstr3d 2652	. . . . 5 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> B C_ ran ( C X. D))
19		rnxpss 4344	. . . . . 6 |- ran ( C X. D) C_ D
20	19	a1i 8	. . . . 5 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> ran ( C X. D) C_ D)
21	18, 20	sstrd 2627	. . . 4 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> B C_ D)
22	11, 21	jca 310	. . 3 |- (((A X. B) =/= (/) /\ (A X. B) C_ (C X. D)) -> (A C_ C /\ B C_ D))
23	22	ex 402	. 2 |- ((A X. B) =/= (/) -> ((A X. B) C_ (C X. D) -> (A C_ C /\ B C_ D)))
24		xpss12 4089	. 2 |- ((A C_ C /\ B C_ D) -> (A X. B) C_ (C X. D))
25	23, 24	impbid1 575	1 |- ((A X. B) =/= (/) -> ((A X. B) C_ (C X. D) <-> (A C_ C /\ B C_ D)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   =/= wne 2017   C_ wss 2593  (/)c0 2875   X. cxp 3984  dom cdm 3986  ran crn 3987

This theorem is referenced by:  xp11 4347

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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