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Theorem xpid11 5067
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  |-  ( ( A  X.  A )  =  ( B  X.  B )  <->  A  =  B )

Proof of Theorem xpid11
StepHypRef Expression
1 dmeq 5046 . . 3  |-  ( ( A  X.  A )  =  ( B  X.  B )  ->  dom  ( A  X.  A
)  =  dom  ( B  X.  B ) )
2 dmxpid 5065 . . 3  |-  dom  ( A  X.  A )  =  A
3 dmxpid 5065 . . 3  |-  dom  ( B  X.  B )  =  B
41, 2, 33eqtr3g 2484 . 2  |-  ( ( A  X.  A )  =  ( B  X.  B )  ->  A  =  B )
5 xpeq12 4864 . . 3  |-  ( ( A  =  B  /\  A  =  B )  ->  ( A  X.  A
)  =  ( B  X.  B ) )
65anidms 649 . 2  |-  ( A  =  B  ->  ( A  X.  A )  =  ( B  X.  B
) )
74, 6impbii 190 1  |-  ( ( A  X.  A )  =  ( B  X.  B )  <->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    X. cxp 4843   dom cdm 4845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-dm 4855
This theorem is referenced by:  intopsn  16450  grporn  25826  resgrprn  25894  ismndo2  25959  rngomndo  26035  rngosn3  26040  ghomgrp  30137  ghomfo  30138
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