Theorem eceq1 6341

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 3461	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 4787	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6308	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6308	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2146	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1290   {csn 3450   "cima 4454   [cec 6304

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4457  df-cnv 4459  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-ec 6308

This theorem is referenced by:  eceq1d  6342  ecelqsg  6359  snec  6367  qliftfun  6388  qliftfuns  6390  qliftval  6392  ecoptocl  6393  eroveu  6397  th3qlem1  6408  th3qlem2  6409  th3q  6411  dmaddpqlem  6990  nqpi  6991  1qec  7001  nqnq0  7054  nq0nn  7055  mulnnnq0  7063  addpinq1  7077  caucvgsrlemfv  7390  caucvgsr  7401  pitonnlem1  7436  axcaucvg  7489