Theorem coeq1 4761

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

Step	Hyp	Ref	Expression
1		coss1 4759	. . 3 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
2		coss1 4759	. . 3 |- ( B C_ A -> ( B o. C ) C_ ( A o. C ) )
3	1, 2	anim12i 336	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( ( A o. C ) C_ ( B o. C ) /\ ( B o. C ) C_ ( A o. C ) ) )
4		eqss 3157	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3157	. 2 |- ( ( A o. C ) = ( B o. C ) <-> ( ( A o. C ) C_ ( B o. C ) /\ ( B o. C ) C_ ( A o. C ) ) )
6	3, 4, 5	3imtr4i 200	1 |- ( A = B -> ( A o. C ) = ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   C_ wss 3116   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by:  coeq1i  4763  coeq1d  4765  coi2  5120  relcnvtr  5123  funcoeqres  5463  ereq1  6508  updjud  7047