Theorem eceq1 6654

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 3643	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 5021	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6621	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6621	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2262	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1372   {csn 3632   "cima 4677   [cec 6617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-ec 6621

This theorem is referenced by:  eceq1d  6655  ecelqsg  6674  snec  6682  qliftfun  6703  qliftfuns  6705  qliftval  6707  ecoptocl  6708  eroveu  6712  th3qlem1  6723  th3qlem2  6724  th3q  6726  dmaddpqlem  7489  nqpi  7490  1qec  7500  nqnq0  7553  nq0nn  7554  mulnnnq0  7562  addpinq1  7576  caucvgsrlemfv  7903  caucvgsr  7914  pitonnlem1  7957  axcaucvg  8012  divsfval  13131  divsfvalg  13132  qusghm  13589  znzrhval  14380