Theorem eceq1 7782

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 4187	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 5466	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 7744	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 7744	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2681	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1483   {csn 4177   "cima 5117   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  eceq1d  7783  ecelqsg  7802  snec  7810  qliftfun  7832  qliftfuns  7834  qliftval  7836  ecoptocl  7837  eroveu  7842  erov  7844  divsfval  16207  qusghm  17697  sylow1lem3  18015  efgi2  18138  frgpup3lem  18190  znzrhval  19895  qustgpopn  21923  qustgplem  21924  elpi1i  22846  pi1xfrf  22853  pi1xfrval  22854  pi1xfrcnvlem  22856  pi1cof  22859  pi1coval  22860  vitalilem3  23379  eceq1i  34039  prtlem9  34149  prtlem11  34151