Theorem eceq1 8545

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 4572	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	imaeq2d 5972	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3		df-ec 8509	. 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴})
4		df-ec 8509	. 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 4562   “ cima 5593  [cec 8505

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 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5076  df-opab 5138  df-xp 5596  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ec 8509

This theorem is referenced by:  eceq1d  8546  ecelqsg  8570  snec  8578  qliftfun  8600  qliftfuns  8602  qliftval  8604  ecoptocl  8605  eroveu  8610  erov  8612  divsfval  17267  qusghm  18880  sylow1lem3  19214  efgi2  19340  frgpup3lem  19392  znzrhval  20763  qustgpopn  23280  qustgplem  23281  elpi1i  24218  pi1xfrf  24225  pi1xfrval  24226  pi1xfrcnvlem  24228  pi1cof  24231  pi1coval  24232  vitalilem3  24783  tgjustr  26844  qusker  31558  qusvscpbl  31560  qusscaval  31561  eceq1i  36419  prtlem9  36885  prtlem11  36887