Theorem eceq1 8738

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 4638	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 6058	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8702	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8702	. 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 4628   “ cima 5679  [cec 8698

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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8702

This theorem is referenced by:  eceq1d  8739  ecelqsg  8763  snec  8771  qliftfun  8793  qliftfuns  8795  qliftval  8797  ecoptocl  8798  eroveu  8803  erov  8805  divsfval  17490  qusghm  19124  sylow1lem3  19463  efgi2  19588  frgpup3lem  19640  znzrhval  21094  qustgpopn  23616  qustgplem  23617  elpi1i  24554  pi1xfrf  24561  pi1xfrval  24562  pi1xfrcnvlem  24564  pi1cof  24567  pi1coval  24568  vitalilem3  25119  tgjustr  27715  qusker  32453  qusvscpbl  32455  qusvsval  32456  eceq1i  37133  disjressuc2  37247  disjlem14  37657  prtlem9  37723  prtlem11  37725  rngqiprngimfv  46764  rngqiprngimf1  46766  rngqiprngimfo  46767