Theorem coeq1 4782

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 4780	. . 3 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
2		coss1 4780	. . 3 |- ( B C_ A -> ( B o. C ) C_ ( A o. C ) )
3	1, 2	anim12i 338	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( ( A o. C ) C_ ( B o. C ) /\ ( B o. C ) C_ ( A o. C ) ) )
4		eqss 3170	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3170	. 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 201	1 |- ( A = B -> ( A o. C ) = ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   C_ wss 3129   o. ccom 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-br 4003  df-opab 4064  df-co 4634

This theorem is referenced by:  coeq1i  4784  coeq1d  4786  coi2  5143  relcnvtr  5146  funcoeqres  5490  ereq1  6538  updjud  7077