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Theorem eceq1 6430
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 3506 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 4849 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 6397 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 6397 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2173 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   {csn 3495   "cima 4510   [cec 6393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-ec 6397
This theorem is referenced by:  eceq1d  6431  ecelqsg  6448  snec  6456  qliftfun  6477  qliftfuns  6479  qliftval  6481  ecoptocl  6482  eroveu  6486  th3qlem1  6497  th3qlem2  6498  th3q  6500  dmaddpqlem  7149  nqpi  7150  1qec  7160  nqnq0  7213  nq0nn  7214  mulnnnq0  7222  addpinq1  7236  caucvgsrlemfv  7563  caucvgsr  7574  pitonnlem1  7617  axcaucvg  7672
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