Theorem eceq1 8494

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 4568	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 5958	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8458	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8458	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2804	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1539  {csn 4558   “ cima 5583  [cec 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ec 8458

This theorem is referenced by:  eceq1d  8495  ecelqsg  8519  snec  8527  qliftfun  8549  qliftfuns  8551  qliftval  8553  ecoptocl  8554  eroveu  8559  erov  8561  divsfval  17175  qusghm  18786  sylow1lem3  19120  efgi2  19246  frgpup3lem  19298  znzrhval  20666  qustgpopn  23179  qustgplem  23180  elpi1i  24115  pi1xfrf  24122  pi1xfrval  24123  pi1xfrcnvlem  24125  pi1cof  24128  pi1coval  24129  vitalilem3  24679  tgjustr  26739  qusker  31451  qusvscpbl  31453  qusscaval  31454  eceq1i  36338  prtlem9  36805  prtlem11  36807