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Theorem coeq1 4521
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 4519 . . 3  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
2 coss1 4519 . . 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 3015 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3015 . 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 1285    C_ wss 2974    o. ccom 4375
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-br 3794  df-opab 3848  df-co 4380
This theorem is referenced by:  coeq1i  4523  coeq1d  4525  coi2  4867  relcnvtr  4870  funcoeqres  5188  ereq1  6179
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