Theorem eceq1 7669

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 4135	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 5385	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 7631	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 7631	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2669	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1475  {csn 4125   “ cima 5041  [cec 7627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631

This theorem is referenced by:  eceq1d  7670  ecelqsg  7689  snec  7697  qliftfun  7719  qliftfuns  7721  qliftval  7723  ecoptocl  7724  eroveu  7729  erov  7731  divsfval  16030  qusghm  17520  sylow1lem3  17838  efgi2  17961  frgpup3lem  18013  znzrhval  19714  qustgpopn  21733  qustgplem  21734  elpi1i  22654  pi1xfrf  22661  pi1xfrval  22662  pi1xfrcnvlem  22664  pi1cof  22667  pi1coval  22668  vitalilem3  23185  prtlem9  33167  prtlem11  33169