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Theorem eceq1 7900
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 4295 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21imaeq2d 5576 . 2 (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3 df-ec 7864 . 2 [𝐴]𝐶 = (𝐶 “ {𝐴})
4 df-ec 7864 . 2 [𝐵]𝐶 = (𝐶 “ {𝐵})
52, 3, 43eqtr4g 2783 1 (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  {csn 4285  cima 5221  [cec 7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ec 7864
This theorem is referenced by:  eceq1d  7901  ecelqsg  7920  snec  7928  qliftfun  7950  qliftfuns  7952  qliftval  7954  ecoptocl  7955  eroveu  7960  erov  7962  divsfval  16330  qusghm  17819  sylow1lem3  18136  efgi2  18259  frgpup3lem  18311  znzrhval  20018  qustgpopn  22045  qustgplem  22046  elpi1i  22967  pi1xfrf  22974  pi1xfrval  22975  pi1xfrcnvlem  22977  pi1cof  22980  pi1coval  22981  vitalilem3  23499  eceq1i  34282  prtlem9  34570  prtlem11  34572
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