Theorem xpss12OLD 4090

Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52.

Assertion

Ref	Expression
xpss12OLD	|- ((A C_ B /\ C C_ D) -> (A X. C) C_ (B X. D))

Proof of Theorem xpss12OLD

Step	Hyp	Ref	Expression
1		relxp 4088	. . 3 |- Rel (A X. C)
2	1	a1i 8	. 2 |- ((A C_ B /\ C C_ D) -> Rel (A X. C))
3		prth 615	. . . 4 |- (((x e. A -> x e. B) /\ (y e. C -> y e. D)) -> ((x e. A /\ y e. C) -> (x e. B /\ y e. D)))
4		visset 2295	. . . . 5 |- y e. _V
5	4	opelxp 4036	. . . 4 |- (<.x, y>. e. (A X. C) <-> (x e. A /\ y e. C))
6	4	opelxp 4036	. . . 4 |- (<.x, y>. e. (B X. D) <-> (x e. B /\ y e. D))
7	3, 5, 6	3imtr4g 612	. . 3 |- (((x e. A -> x e. B) /\ (y e. C -> y e. D)) -> (<.x, y>. e. (A X. C) -> <.x, y>. e. (B X. D)))
8		ssel 2615	. . 3 |- (A C_ B -> (x e. A -> x e. B))
9		ssel 2615	. . 3 |- (C C_ D -> (y e. C -> y e. D))
10	7, 8, 9	syl2an 503	. 2 |- ((A C_ B /\ C C_ D) -> (<.x, y>. e. (A X. C) -> <.x, y>. e. (B X. D)))
11	2, 10	relssdv 4079	1 |- ((A C_ B /\ C C_ D) -> (A X. C) C_ (B X. D))

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300   C_ wss 2593  <.cop 3046   X. cxp 3984  Rel wrel 3991

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-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-opab 3396  df-xp 4000  df-rel 4001

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