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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
StepHypRef Expression
1 dmeq 5208 . . 3
2 dmxpid 5227 . . 3
3 dmxpid 5227 . . 3
41, 2, 33eqtr3g 2521 . 2
5 xpeq12 5023 . . 3
65anidms 645 . 2
74, 6impbii 188 1
 Colors of variables: wff setvar class 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
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