Theorem coeq2 4913

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 4911	. . 3 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )
2		coss2 4911	. . 3 |- ( B C_ A -> ( C o. B ) C_ ( C o. A ) )
3	1, 2	anim12i 338	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( ( C o. A ) C_ ( C o. B ) /\ ( C o. B ) C_ ( C o. A ) ) )
4		eqss 3253	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3253	. 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 201	1 |- ( A = B -> ( C o. A ) = ( C o. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   C_ wss 3211   o. ccom 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-br 4110  df-opab 4172  df-co 4758

This theorem is referenced by:  coeq2i  4915  coeq2d  4917  coi2  5279  relcnvtr  5282  relcoi1  5294  f1eqcocnv  5964  ereq1  6774  seqf1oglem2  10882  seqf1og  10883  gsumwmhm  13711  upxp  15137  uptx  15139  txcn  15140  gfsumval  16862