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Theorem eceq1 7347
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 4037 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 5337 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 7313 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 7313 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2533 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   {csn 4027   "cima 5002   [cec 7309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-ec 7313
This theorem is referenced by:  eceq1d  7348  ecelqsg  7366  snec  7374  qliftfun  7396  qliftfuns  7398  qliftval  7400  ecoptocl  7401  eroveu  7406  erov  7408  divsfval  14802  divsghm  16108  sylow1lem3  16426  efgi2  16549  frgpup3lem  16601  znzrhval  18380  divstgpopn  20381  divstgplem  20382  elpi1i  21309  pi1xfrf  21316  pi1xfrval  21317  pi1xfrcnvlem  21319  pi1cof  21322  pi1coval  21323  vitalilem3  21782  ismntoplly  27671  prtlem9  30237  prtlem11  30239
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