Theorem eceq1 8330

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 4580	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 5932	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8294	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8294	. 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵})
5	2, 3, 4	3eqtr4g 2884	1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Syntax hints:   → wi 4   = wceq 1536  {csn 4570   “ cima 5561  [cec 8290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ec 8294

This theorem is referenced by:  eceq1d  8331  ecelqsg  8355  snec  8363  qliftfun  8385  qliftfuns  8387  qliftval  8389  ecoptocl  8390  eroveu  8395  erov  8397  divsfval  16823  qusghm  18398  sylow1lem3  18728  efgi2  18854  frgpup3lem  18906  znzrhval  20696  qustgpopn  22731  qustgplem  22732  elpi1i  23653  pi1xfrf  23660  pi1xfrval  23661  pi1xfrcnvlem  23663  pi1cof  23666  pi1coval  23667  vitalilem3  24214  tgjustr  26263  qusker  30922  qusvscpbl  30924  qusscaval  30925  eceq1i  35537  prtlem9  36004  prtlem11  36006