Theorem eceq1 8802

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 4658	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6089	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8765	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8765	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2805	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1537  {csn 4648   “ cima 5703  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765

This theorem is referenced by:  eceq1d  8803  ecelqsg  8830  snec  8838  qliftfun  8860  qliftfuns  8862  qliftval  8864  ecoptocl  8865  eroveu  8870  erov  8872  divsfval  17607  qusghm  19295  sylow1lem3  19642  efgi2  19767  frgpup3lem  19819  rngqiprngimfv  21331  rngqiprngimf1  21333  rngqiprngimfo  21334  pzriprnglem11  21525  znzrhval  21588  qustgpopn  24149  qustgplem  24150  elpi1i  25098  pi1xfrf  25105  pi1xfrval  25106  pi1xfrcnvlem  25108  pi1cof  25111  pi1coval  25112  vitalilem3  25664  tgjustr  28500  qusker  33342  qusvscpbl  33344  qusvsval  33345  algextdeg  33716  eceq1i  38232  disjressuc2  38344  disjlem14  38754  prtlem9  38820  prtlem11  38822  aks6d1c6lem5  42134  aks5lem3a  42146