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Theorem coeq2 4993
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
StepHypRef Expression
1 coss2 4991 . . 3  |-  ( A 
C_  B  ->  ( C  o.  A )  C_  ( C  o.  B
) )
2 coss2 4991 . . 3  |-  ( B 
C_  A  ->  ( C  o.  B )  C_  ( C  o.  A
) )
31, 2anim12i 570 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( C  o.  A )  C_  ( C  o.  B )  /\  ( C  o.  B
)  C_  ( C  o.  A ) ) )
4 eqss 3447 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3447 . 2  |-  ( ( C  o.  A )  =  ( C  o.  B )  <->  ( ( C  o.  A )  C_  ( C  o.  B
)  /\  ( C  o.  B )  C_  ( C  o.  A )
) )
63, 4, 53imtr4i 270 1  |-  ( A  =  B  ->  ( C  o.  A )  =  ( C  o.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    C_ wss 3404    o. ccom 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-in 3411  df-ss 3418  df-br 4403  df-opab 4462  df-co 4843
This theorem is referenced by:  coeq2i  4995  coeq2d  4997  coi2  5352  relcnvtr  5355  relcoi1OLD  5365  f1eqcocnv  6199  ereq1  7370  seqf1olem2  12253  seqf1o  12254  relexpsucnnr  13088  isps  16448  pwsco2mhm  16618  gsumwmhm  16629  frmdgsum  16646  frmdup1  16648  frmdup2  16649  symgov  17031  pmtr3ncom  17116  psgnunilem1  17134  frgpuplem  17422  frgpupf  17423  frgpupval  17424  gsumval3eu  17538  gsumval3lem2  17540  kgencn2  20572  upxp  20638  uptx  20640  txcn  20641  xkococnlem  20674  xkococn  20675  cnmptk1  20696  cnmptkk  20698  xkofvcn  20699  imasdsf1olem  21388  pi1cof  22090  pi1coval  22091  elovolmr  22429  ovoliunlem3  22457  ismbf1  22582  motplusg  24587  hocsubdir  27438  hoddi  27643  lnopco0i  27657  opsqrlem1  27793  pjsdi2i  27810  pjin2i  27846  pjclem1  27848  symgfcoeu  28608  eulerpartgbij  29205  cvmliftmo  30007  cvmliftlem14  30020  cvmliftiota  30024  cvmlift2lem13  30038  cvmlift2  30039  cvmliftphtlem  30040  cvmlift3lem2  30043  cvmlift3lem6  30047  cvmlift3lem7  30048  cvmlift3lem9  30050  cvmlift3  30051  msubco  30169  ftc1anclem8  32024  upixp  32056  trlcoat  34290  trljco  34307  tgrpov  34315  tendovalco  34332  erngmul  34373  erngmul-rN  34381  dvamulr  34579  dvavadd  34582  dvhmulr  34654  dihjatcclem4  34989  mendmulr  36054  hoiprodcl2  38377  ovnlecvr  38380  ovncvrrp  38386  ovnsubaddlem2  38393  ovncvr2  38433  opnvonmbllem1  38454  opnvonmbl  38456  rngcinv  40036  rngcinvALTV  40048  ringcinv  40087  ringcinvALTV  40111
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