Theorem dmxpss 3473

Description: The domain of a cross product is a subclass of the first factor.

Assertion

Ref	Expression
dmxpss	|- dom ( A X. B) (_ A

Proof of Theorem dmxpss

Step	Hyp	Ref	Expression
1		0ss 2301	. . 3 |- (/) (_ A
2		xpeq2 3201	. . . . . . 7 |- (B = (/) -> (A X. B) = (A X. (/)))
3		xp0 3465	. . . . . . 7 |- (A X. (/)) = (/)
4	2, 3	syl6eq 1523	. . . . . 6 |- (B = (/) -> (A X. B) = (/))
5	4	dmeqd 3313	. . . . 5 |- (B = (/) -> dom ( A X. B) = dom (/))
6		dm0 3323	. . . . 5 |- dom (/) = (/)
7	5, 6	syl6eq 1523	. . . 4 |- (B = (/) -> dom ( A X. B) = (/))
8	7	sseq1d 2088	. . 3 |- (B = (/) -> (dom ( A X. B) (_ A <-> (/) (_ A))
9	1, 8	mpbiri 194	. 2 |- (B = (/) -> dom ( A X. B) (_ A)
10		dmxp 3332	. . 3 |- (B =/= (/) -> dom ( A X. B) = A)
11		eqimss 2109	. . 3 |- (dom ( A X. B) = A -> dom ( A X. B) (_ A)
12	10, 11	syl 10	. 2 |- (B =/= (/) -> dom ( A X. B) (_ A)
13	9, 12	pm2.61ine 1634	1 |- dom ( A X. B) (_ A

Syntax hints:   = wceq 956   =/= wne 1585   (_ wss 2047  (/)c0 2280   X. cxp 3168  dom cdm 3170

This theorem is referenced by:  ssxpr 3475  funssxp 3638  dff2 3817  brdom3 4801  brdom5 4802  brdom4 4803

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188

Copyright terms: Public domain