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Theorem xpid11 5224
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 5203 . . 3  |-  ( ( A  X.  A )  =  ( B  X.  B )  ->  dom  ( A  X.  A
)  =  dom  ( B  X.  B ) )
2 dmxpid 5222 . . 3  |-  dom  ( A  X.  A )  =  A
3 dmxpid 5222 . . 3  |-  dom  ( B  X.  B )  =  B
41, 2, 33eqtr3g 2531 . 2  |-  ( ( A  X.  A )  =  ( B  X.  B )  ->  A  =  B )
5 xpeq12 5018 . . 3  |-  ( ( A  =  B  /\  A  =  B )  ->  ( A  X.  A
)  =  ( B  X.  B ) )
65anidms 645 . 2  |-  ( A  =  B  ->  ( A  X.  A )  =  ( B  X.  B
) )
74, 6impbii 188 1  |-  ( ( A  X.  A )  =  ( B  X.  B )  <->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    X. cxp 4997   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009
This theorem is referenced by:  intopsn  15785  grporn  24987  resgrprn  25055  ismndo2  25120  rngomndo  25196  rngosn3  25201  ghomgrp  28781  ghomfo  28782
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