Theorem xpid11 5229

Description: The Cartesian product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
xpid11

Proof of Theorem xpid11

Step	Hyp	Ref	Expression
1		dmeq 5208	. . 3
2		dmxpid 5227	. . 3
3		dmxpid 5227	. . 3
4	1, 2, 3	3eqtr3g 2521	. 2
5		xpeq12 5023	. . 3
6	5	anidms 645	. 2
7	4, 6	impbii 188	1

Syntax hints:  <->wb 184  =wceq 1395  X.cxp 5002  domcdm 5004

This theorem is referenced by:  intopsn  15882  grporn  25214  resgrprn  25282  ismndo2  25347  rngomndo  25423  rngosn3  25428  ghomgrp  29030  ghomfo  29031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-dm 5014