Theorem dmxpss 3465

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 2297	. . 3 |- (/) (_ A
2		xpeq2 3196	. . . . . . 7 |- (B = (/) -> (A X. B) = (A X. (/)))
3		xp0 3457	. . . . . . 7 |- (A X. (/)) = (/)
4	2, 3	syl6eq 1520	. . . . . 6 |- (B = (/) -> (A X. B) = (/))
5	4	dmeqd 3308	. . . . 5 |- (B = (/) -> dom ( A X. B) = dom (/))
6		dm0 3318	. . . . 5 |- dom (/) = (/)
7	5, 6	syl6eq 1520	. . . 4 |- (B = (/) -> dom ( A X. B) = (/))
8	7	sseq1d 2084	. . 3 |- (B = (/) -> (dom ( A X. B) (_ A <-> (/) (_ A))
9	1, 8	mpbiri 194	. 2 |- (B = (/) -> dom ( A X. B) (_ A)
10		dmxp 3327	. . 3 |- (B =/= (/) -> dom ( A X. B) = A)
11		eqimss 2105	. . 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 1631	1 |- dom ( A X. B) (_ A

Syntax hints:   = wceq 954   =/= wne 1582   (_ wss 2043  (/)c0 2276   X. cxp 3163  dom cdm 3165

This theorem is referenced by:  ssxpr 3467  funssxp 3629  dff2 3808  brdom3 4781  brdom5 4782  brdom4 4783

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183

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