Theorem xpid11 3341

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 3317	. . 3 |- ((A X. A) = (B X. B) -> dom ( A X. A) = dom ( B X. B))
2		dmxpid 3339	. . 3 |- dom ( A X. A) = A
3		dmxpid 3339	. . 3 |- dom ( B X. B) = B
4	1, 2, 3	3eqtr3g 1533	. 2 |- ((A X. A) = (B X. B) -> A = B)
5		xpeq1 3206	. . 3 |- (A = B -> (A X. A) = (B X. A))
6		xpeq2 3207	. . 3 |- (A = B -> (B X. A) = (B X. B))
7	5, 6	eqtrd 1510	. 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 958   X. cxp 3174  dom cdm 3176

This theorem is referenced by:  grprn 8053  resgrprn 8091  ghomgrp 10385  ghomfo 10386  isalg 10624  algi 10631

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-dm 3194

Copyright terms: Public domain