Theorem rnxpssOLD 4345

Description: The range of a cross product is a subclass of the second factor.

Assertion

Ref	Expression
rnxpssOLD	|- ran ( A X. B) C_ B

Proof of Theorem rnxpssOLD

Step	Hyp	Ref	Expression
1		0ss 2900	. . 3 |- (/) C_ B
2		xpeq1 4016	. . . . . . 7 |- (A = (/) -> (A X. B) = ((/) X. B))
3		xp0r 4065	. . . . . . 7 |- ((/) X. B) = (/)
4	2, 3	syl6eq 1944	. . . . . 6 |- (A = (/) -> (A X. B) = (/))
5	4	rneqd 4188	. . . . 5 |- (A = (/) -> ran ( A X. B) = ran (/))
6		rn0 4203	. . . . 5 |- ran (/) = (/)
7	5, 6	syl6eq 1944	. . . 4 |- (A = (/) -> ran ( A X. B) = (/))
8	7	sseq1d 2644	. . 3 |- (A = (/) -> (ran ( A X. B) C_ B <-> (/) C_ B))
9	1, 8	mpbiri 211	. 2 |- (A = (/) -> ran ( A X. B) C_ B)
10		rnxp 4342	. . 3 |- (A =/= (/) -> ran ( A X. B) = B)
11		eqimss 2665	. . 3 |- (ran ( A X. B) = B -> ran ( A X. B) C_ B)
12	10, 11	syl 12	. 2 |- (A =/= (/) -> ran ( A X. B) C_ B)
13	9, 12	pm2.61ine 2089	1 |- ran ( A X. B) C_ B

Syntax hints:   = wceq 1298   =/= wne 2017   C_ wss 2593  (/)c0 2875   X. cxp 3984  ran crn 3987

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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