Theorem eceq1 8661

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 4586	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6009	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8624	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8624	. 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 4576   “ cima 5619  [cec 8620

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 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ec 8624

This theorem is referenced by:  eceq1d  8662  ecelqs  8692  snec  8702  qliftfun  8726  qliftfuns  8728  qliftval  8730  ecoptocl  8731  eroveu  8736  erov  8738  divsfval  17451  qusghm  19168  sylow1lem3  19513  efgi2  19638  frgpup3lem  19690  rngqiprngimfv  21236  rngqiprngimf1  21238  rngqiprngimfo  21239  pzriprnglem11  21429  znzrhval  21484  qustgpopn  24036  qustgplem  24037  elpi1i  24974  pi1xfrf  24981  pi1xfrval  24982  pi1xfrcnvlem  24984  pi1cof  24987  pi1coval  24988  vitalilem3  25539  tgjustr  28453  qusker  33312  qusvscpbl  33314  qusvsval  33315  algextdeg  33736  eceq1i  38318  disjressuc2  38426  disjlem14  38842  prtlem9  38909  prtlem11  38911  aks6d1c6lem5  42216  aks5lem3a  42228