Theorem coeq1 4853

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 4851	. . 3 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
2		coss1 4851	. . 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 3216	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3216	. 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 1373   C_ wss 3174   o. ccom 4697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-br 4060  df-opab 4122  df-co 4702

This theorem is referenced by:  coeq1i  4855  coeq1d  4857  coi2  5218  relcnvtr  5221  funcoeqres  5575  ereq1  6650  updjud  7210  seqf1oglem2  10702  seqf1og  10703