Theorem coeq2 4886

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 4884	. . 3 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )
2		coss2 4884	. . 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 3240	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3240	. 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 1395   C_ wss 3198   o. ccom 4727

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3204  df-ss 3211  df-br 4087  df-opab 4149  df-co 4732

This theorem is referenced by:  coeq2i  4888  coeq2d  4890  coi2  5251  relcnvtr  5254  relcoi1  5266  f1eqcocnv  5927  ereq1  6704  seqf1oglem2  10772  seqf1og  10773  gsumwmhm  13571  upxp  14986  uptx  14988  txcn  14989