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Theorem coeq1 4766
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 4764 . . 3  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
2 coss1 4764 . . 3  |-  ( B 
C_  A  ->  ( B  o.  C )  C_  ( A  o.  C
) )
31, 2anim12i 336 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( A  o.  C )  C_  ( B  o.  C )  /\  ( B  o.  C
)  C_  ( A  o.  C ) ) )
4 eqss 3162 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3162 . 2  |-  ( ( A  o.  C )  =  ( B  o.  C )  <->  ( ( A  o.  C )  C_  ( B  o.  C
)  /\  ( B  o.  C )  C_  ( A  o.  C )
) )
63, 4, 53imtr4i 200 1  |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    C_ wss 3121    o. ccom 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3988  df-opab 4049  df-co 4618
This theorem is referenced by:  coeq1i  4768  coeq1d  4770  coi2  5125  relcnvtr  5128  funcoeqres  5471  ereq1  6516  updjud  7055
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