Theorem coeq2 4437

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 4435	. . 3 |- (A C_ B -> ( C o. A) C_ ( C o. B))
2		coss2 4435	. . 3 |- (B C_ A -> ( C o. B) C_ ( C o. A))
3	1, 2	anim12i 321	. 2 |- ((A C_ B /\ B C_ A) -> (( C o. A) C_ ( C o. B) /\ ( C o. B) C_ ( C o. A)))
4		eqss 2954	. 2 |- (A = B <-> (A C_ B /\ B C_ A))
5		eqss 2954	. 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 190	1 |- (A = B -> ( C o. A) = ( C o. B))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   C_ wss 2911   o. ccom 4292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-co 4297

This theorem is referenced by:  coeq2i  4439  coeq2d  4441  coi2  4780  relcnvtr  4783  relcoi1  4792  f1eqcocnv  5374  ereq1  6049