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Theorem eceq1 6229
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
eceq1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )

Proof of Theorem eceq1
StepHypRef Expression
1 sneq 3428 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 4719 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 6196 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 6196 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2140 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   {csn 3417   "cima 4395   [cec 6192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-cnv 4400  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-ec 6196
This theorem is referenced by:  eceq1d  6230  ecelqsg  6247  snec  6255  qliftfun  6276  qliftfuns  6278  qliftval  6280  ecoptocl  6281  eroveu  6285  th3qlem1  6296  th3qlem2  6297  th3q  6299  dmaddpqlem  6665  nqpi  6666  1qec  6676  nqnq0  6729  nq0nn  6730  mulnnnq0  6738  addpinq1  6752  caucvgsrlemfv  7065  caucvgsr  7076  pitonnlem1  7111  axcaucvg  7164
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