Theorem eceq1 8758

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 4611	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6047	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8721	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8721	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2795	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1540  {csn 4601   “ cima 5657  [cec 8717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ec 8721

This theorem is referenced by:  eceq1d  8759  ecelqsg  8786  snec  8794  qliftfun  8816  qliftfuns  8818  qliftval  8820  ecoptocl  8821  eroveu  8826  erov  8828  divsfval  17561  qusghm  19238  sylow1lem3  19581  efgi2  19706  frgpup3lem  19758  rngqiprngimfv  21259  rngqiprngimf1  21261  rngqiprngimfo  21262  pzriprnglem11  21452  znzrhval  21507  qustgpopn  24058  qustgplem  24059  elpi1i  24997  pi1xfrf  25004  pi1xfrval  25005  pi1xfrcnvlem  25007  pi1cof  25010  pi1coval  25011  vitalilem3  25563  tgjustr  28453  qusker  33364  qusvscpbl  33366  qusvsval  33367  algextdeg  33759  eceq1i  38294  disjressuc2  38406  disjlem14  38816  prtlem9  38882  prtlem11  38884  aks6d1c6lem5  42190  aks5lem3a  42202