Theorem eceq1 8680

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 4572	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6019	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8642	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8642	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2800	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1547  {csn 4562   “ cima 5628  [cec 8638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8642

This theorem is referenced by:  eceq1d  8681  ecelqs  8711  snecg  8721  snec  8722  qliftfun  8746  qliftfuns  8748  qliftval  8750  ecoptocl  8751  eroveu  8756  erov  8758  divsfval  17509  qusghm  19228  sylow1lem3  19573  efgi2  19698  frgpup3lem  19750  rngqiprngimfv  21298  rngqiprngimf1  21300  rngqiprngimfo  21301  pzriprnglem11  21473  znzrhval  21528  qustgpopn  24110  qustgplem  24111  elpi1i  25038  pi1xfrf  25045  pi1xfrval  25046  pi1xfrcnvlem  25048  pi1cof  25051  pi1coval  25052  vitalilem3  25602  tgjustr  28567  qusker  33439  qusvscpbl  33441  qusvsval  33442  algextdeg  33916  eceq1i  38658  disjressuc2  38785  ecqmap  38823  disjimeceqim2  39179  disjimeceqbi  39180  disjimeceqbi2  39181  disjimrmoeqec  39182  qmapeldisjsbi  39235  disjlem14  39275  prtlem9  39363  prtlem11  39365  aks6d1c6lem5  42669  aks5lem3a  42681