Theorem dmxpin 3334

Description: The domain of the intersection of two square cross products. Unlike dmin 3318, equality holds.

Assertion

Ref	Expression
dmxpin	|- dom ((A X. A) i^i (B X. B)) = (A i^i B)

Proof of Theorem dmxpin

Step	Hyp	Ref	Expression
1		inxp 3269	. . 3 |- ((A X. A) i^i (B X. B)) = ((A i^i B) X. (A i^i B))
2	1	dmeqi 3312	. 2 |- dom ((A X. A) i^i (B X. B)) = dom ((A i^i B) X. (A i^i B))
3		dmxpid 3333	. 2 |- dom ((A i^i B) X. (A i^i B)) = (A i^i B)
4	2, 3	eqtr 1495	1 |- dom ((A X. A) i^i (B X. B)) = (A i^i B)

Syntax hints:   = wceq 956   i^i cin 2046   X. cxp 3168  dom cdm 3170

This theorem is referenced by:  metssba 7809

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-dm 3188

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