Theorem coeq2 4697

Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Assertion

Ref	Expression
coeq2	|- ( A = B -> ( C o. A ) = ( C o. B ) )

Proof of Theorem coeq2

Step	Hyp	Ref	Expression
1		coss2 4695	. . 3 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )
2		coss2 4695	. . 3 |- ( B C_ A -> ( C o. B ) C_ ( C o. A ) )
3	1, 2	anim12i 336	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( ( C o. A ) C_ ( C o. B ) /\ ( C o. B ) C_ ( C o. A ) ) )
4		eqss 3112	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3112	. 2 |- ( ( C o. A ) = ( C o. B ) <-> ( ( C o. A ) C_ ( C o. B ) /\ ( C o. B ) C_ ( C o. A ) ) )
6	3, 4, 5	3imtr4i 200	1 |- ( A = B -> ( C o. A ) = ( C o. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   C_ wss 3071   o. ccom 4543

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-br 3930  df-opab 3990  df-co 4548

This theorem is referenced by:  coeq2i  4699  coeq2d  4701  coi2  5055  relcnvtr  5058  relcoi1  5070  f1eqcocnv  5692  ereq1  6436  upxp  12441  uptx  12443  txcn  12444