Theorem xpid11 3335

Description: The cross product of a class with itself is one-to-one.

Assertion

Ref	Expression
xpid11	|- ((A X. A) = (B X. B) <-> A = B)

Proof of Theorem xpid11

Step	Hyp	Ref	Expression
1		dmeq 3311	. . 3 |- ((A X. A) = (B X. B) -> dom ( A X. A) = dom ( B X. B))
2		dmxpid 3333	. . 3 |- dom ( A X. A) = A
3		dmxpid 3333	. . 3 |- dom ( B X. B) = B
4	1, 2, 3	3eqtr3g 1530	. 2 |- ((A X. A) = (B X. B) -> A = B)
5		xpeq1 3200	. . 3 |- (A = B -> (A X. A) = (B X. A))
6		xpeq2 3201	. . 3 |- (A = B -> (B X. A) = (B X. B))
7	5, 6	eqtrd 1507	. 2 |- (A = B -> (A X. A) = (B X. B))
8	4, 7	impbi 157	1 |- ((A X. A) = (B X. B) <-> A = B)

Syntax hints:   <-> wb 146   = wceq 956   X. cxp 3168  dom cdm 3170

This theorem is referenced by:  grprn 8056  resgrprn 8095  ghomgrp 10390  ghomfo 10391  isalg 10653  algi 10660

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-dm 3188

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