Theorem eceq1 8667

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

Step	Hyp	Ref	Expression
1		sneq 4585	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6014	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8630	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8630	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2791	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1541  {csn 4575   “ cima 5622  [cec 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8630

This theorem is referenced by:  eceq1d  8668  ecelqs  8698  snec  8708  qliftfun  8732  qliftfuns  8734  qliftval  8736  ecoptocl  8737  eroveu  8742  erov  8744  divsfval  17457  qusghm  19173  sylow1lem3  19518  efgi2  19643  frgpup3lem  19695  rngqiprngimfv  21241  rngqiprngimf1  21243  rngqiprngimfo  21244  pzriprnglem11  21434  znzrhval  21489  qustgpopn  24041  qustgplem  24042  elpi1i  24979  pi1xfrf  24986  pi1xfrval  24987  pi1xfrcnvlem  24989  pi1cof  24992  pi1coval  24993  vitalilem3  25544  tgjustr  28458  qusker  33321  qusvscpbl  33323  qusvsval  33324  algextdeg  33745  eceq1i  38322  disjressuc2  38441  disjlem14  38902  prtlem9  38969  prtlem11  38971  aks6d1c6lem5  42276  aks5lem3a  42288