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Theorem eceq1 6077
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 3378 . . 3 (A = B → {A} = {B})
21imaeq2d 4611 . 2 (A = B → (𝐶 “ {A}) = (𝐶 “ {B}))
3 df-ec 6044 . 2 [A]𝐶 = (𝐶 “ {A})
4 df-ec 6044 . 2 [B]𝐶 = (𝐶 “ {B})
52, 3, 43eqtr4g 2094 1 (A = B → [A]𝐶 = [B]𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {csn 3367  cima 4291  [cec 6040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044
This theorem is referenced by:  eceq1d  6078  ecelqsg  6095  snec  6103  qliftfun  6124  qliftfuns  6126  qliftval  6128  ecoptocl  6129  eroveu  6133  th3qlem1  6144  th3qlem2  6145  th3q  6147  dmaddpqlem  6361  nqpi  6362  1qec  6372  nqnq0  6424  nq0nn  6425  mulnnnq0  6433  addpinq1  6447  pitonnlem1  6741
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