Theorem eceq1 8735

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 4600	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6064	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8697	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8697	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1570  {csn 4590   “ cima 5666  [cec 8693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8697

This theorem is referenced by:  eceq1d  8736  ecelqs  8766  snecg  8776  snec  8777  qliftfun  8801  qliftfuns  8803  qliftval  8805  ecoptocl  8806  eroveu  8811  erov  8813  divsfval  17602  qusghm  19326  sylow1lem3  19671  efgi2  19796  frgpup3lem  19848  rngqiprngimfv  21419  rngqiprngimf1  21421  rngqiprngimfo  21422  pzriprnglem11  21622  znzrhval  21677  qustgpopn  24258  qustgplem  24259  elpi1i  25186  pi1xfrf  25193  pi1xfrval  25194  pi1xfrcnvlem  25196  pi1cof  25199  pi1coval  25200  vitalilem3  25750  tgjustr  28724  qusker  33650  qusvscpbl  33652  qusvsval  33653  algextdeg  34096  eceq1i  38914  disjressuc2  39041  ecqmap  39079  disjimeceqim2  39435  disjimeceqbi  39436  disjimeceqbi2  39437  disjimrmoeqec  39438  qmapeldisjsbi  39491  disjlem14  39531  prtlem9  39619  prtlem11  39621  aks6d1c6lem5  42925  aks5lem3a  42937