Theorem coeq2 4820

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 4818	. . 3 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )
2		coss2 4818	. . 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 3194	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3194	. 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 1364   C_ wss 3153   o. ccom 4663

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166  df-br 4030  df-opab 4091  df-co 4668

This theorem is referenced by:  coeq2i  4822  coeq2d  4824  coi2  5182  relcnvtr  5185  relcoi1  5197  f1eqcocnv  5834  ereq1  6594  seqf1oglem2  10591  seqf1og  10592  gsumwmhm  13070  upxp  14440  uptx  14442  txcn  14443