Theorem eceq1 6570

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 3604	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 4971	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6537	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6537	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2235	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1353   {csn 3593   "cima 4630   [cec 6533

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-ec 6537

This theorem is referenced by:  eceq1d  6571  ecelqsg  6588  snec  6596  qliftfun  6617  qliftfuns  6619  qliftval  6621  ecoptocl  6622  eroveu  6626  th3qlem1  6637  th3qlem2  6638  th3q  6640  dmaddpqlem  7376  nqpi  7377  1qec  7387  nqnq0  7440  nq0nn  7441  mulnnnq0  7449  addpinq1  7463  caucvgsrlemfv  7790  caucvgsr  7801  pitonnlem1  7844  axcaucvg  7899  divsfvalg  12748