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Theorem coeq1 4593
Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
Assertion
Ref Expression
coeq1  |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )

Proof of Theorem coeq1
StepHypRef Expression
1 coss1 4591 . . 3  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
2 coss1 4591 . . 3  |-  ( B 
C_  A  ->  ( B  o.  C )  C_  ( A  o.  C
) )
31, 2anim12i 331 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( A  o.  C )  C_  ( B  o.  C )  /\  ( B  o.  C
)  C_  ( A  o.  C ) ) )
4 eqss 3040 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3040 . 2  |-  ( ( A  o.  C )  =  ( B  o.  C )  <->  ( ( A  o.  C )  C_  ( B  o.  C
)  /\  ( B  o.  C )  C_  ( A  o.  C )
) )
63, 4, 53imtr4i 199 1  |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    C_ wss 2999    o. ccom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-br 3846  df-opab 3900  df-co 4447
This theorem is referenced by:  coeq1i  4595  coeq1d  4597  coi2  4947  relcnvtr  4950  funcoeqres  5284  ereq1  6297  updjud  6771
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