Theorem coeq1 4511

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 4509	. . 3 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
2		coss1 4509	. . 3 |- ( B C_ A -> ( B o. C ) C_ ( A o. C ) )
3	1, 2	anim12i 331	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( ( A o. C ) C_ ( B o. C ) /\ ( B o. C ) C_ ( A o. C ) ) )
4		eqss 3014	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3014	. 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 199	1 |- ( A = B -> ( A o. C ) = ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   o. ccom 4367

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-co 4372

This theorem is referenced by:  coeq1i  4513  coeq1d  4515  coi2  4857  relcnvtr  4860  funcoeqres  5177  ereq1  6136