ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eceq1 GIF version

Theorem eceq1 6207
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
StepHypRef Expression
1 sneq 3417 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21imaeq2d 4698 . 2 (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵}))
3 df-ec 6174 . 2 [𝐴]𝐶 = (𝐶 “ {𝐴})
4 df-ec 6174 . 2 [𝐵]𝐶 = (𝐶 “ {𝐵})
52, 3, 43eqtr4g 2139 1 (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  {csn 3406  cima 4374  [cec 6170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-ec 6174
This theorem is referenced by:  eceq1d  6208  ecelqsg  6225  snec  6233  qliftfun  6254  qliftfuns  6256  qliftval  6258  ecoptocl  6259  eroveu  6263  th3qlem1  6274  th3qlem2  6275  th3q  6277  dmaddpqlem  6629  nqpi  6630  1qec  6640  nqnq0  6693  nq0nn  6694  mulnnnq0  6702  addpinq1  6716  caucvgsrlemfv  7029  caucvgsr  7040  pitonnlem1  7075  axcaucvg  7128
  Copyright terms: Public domain W3C validator