ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  coeq1 Unicode version

Theorem coeq1 4436
Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
Assertion
Ref Expression
coeq1  o.  C  o.  C

Proof of Theorem coeq1
StepHypRef Expression
1 coss1 4434 . . 3 
C_  o.  C  C_  o.  C
2 coss1 4434 . . 3 
C_  o.  C  C_  o.  C
31, 2anim12i 321 . 2  C_  C_  o.  C  C_  o.  C  o.  C  C_  o.  C
4 eqss 2954 . 2 
C_  C_
5 eqss 2954 . 2  o.  C  o.  C  o.  C  C_  o.  C  o.  C  C_  o.  C
63, 4, 53imtr4i 190 1  o.  C  o.  C
Colors of variables: wff set class
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:  coeq1i  4438  coeq1d  4440  coi2  4780  relcnvtr  4783  funcoeqres  5100  ereq1  6049
  Copyright terms: Public domain W3C validator