Theorem eceq1 8674

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 4590	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6019	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8637	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8637	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1541  {csn 4580   “ cima 5627  [cec 8633

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 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by:  eceq1d  8675  ecelqs  8705  snec  8715  qliftfun  8739  qliftfuns  8741  qliftval  8743  ecoptocl  8744  eroveu  8749  erov  8751  divsfval  17468  qusghm  19184  sylow1lem3  19529  efgi2  19654  frgpup3lem  19706  rngqiprngimfv  21253  rngqiprngimf1  21255  rngqiprngimfo  21256  pzriprnglem11  21446  znzrhval  21501  qustgpopn  24064  qustgplem  24065  elpi1i  25002  pi1xfrf  25009  pi1xfrval  25010  pi1xfrcnvlem  25012  pi1cof  25015  pi1coval  25016  vitalilem3  25567  tgjustr  28546  qusker  33430  qusvscpbl  33432  qusvsval  33433  algextdeg  33882  eceq1i  38473  disjressuc2  38592  disjlem14  39053  prtlem9  39120  prtlem11  39122  aks6d1c6lem5  42427  aks5lem3a  42439