Theorem eceq1 8741

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 4639	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6060	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8705	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8705	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2798	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1542  {csn 4629   “ cima 5680  [cec 8701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705

This theorem is referenced by:  eceq1d  8742  ecelqsg  8766  snec  8774  qliftfun  8796  qliftfuns  8798  qliftval  8800  ecoptocl  8801  eroveu  8806  erov  8808  divsfval  17493  qusghm  19129  sylow1lem3  19468  efgi2  19593  frgpup3lem  19645  znzrhval  21102  qustgpopn  23624  qustgplem  23625  elpi1i  24562  pi1xfrf  24569  pi1xfrval  24570  pi1xfrcnvlem  24572  pi1cof  24575  pi1coval  24576  vitalilem3  25127  tgjustr  27725  qusker  32464  qusvscpbl  32466  qusvsval  32467  eceq1i  37144  disjressuc2  37258  disjlem14  37668  prtlem9  37734  prtlem11  37736  rngqiprngimfv  46783  rngqiprngimf1  46785  rngqiprngimfo  46786  pzriprnglem11  46815