Theorem snec 7810

Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
snec.1	|- A e. _V

Assertion

Ref	Expression
snec	|- { [ A ] R } = ( { A } /. R )

Proof of Theorem snec

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snec.1	. . . 4 |- A e. _V
2		eceq1 7782	. . . . 5 |- ( x = A -> [ x ] R = [ A ] R )
3	2	eqeq2d 2632	. . . 4 |- ( x = A -> ( y = [ x ] R <-> y = [ A ] R ) )
4	1, 3	rexsn 4223	. . 3 |- ( E. x e. { A } y = [ x ] R <-> y = [ A ] R )
5	4	abbii 2739	. 2 |- { y | E. x e. { A } y = [ x ] R } = { y | y = [ A ] R }
6		df-qs 7748	. 2 |- ( { A } /. R ) = { y | E. x e. { A } y = [ x ] R }
7		df-sn 4178	. 2 |- { [ A ] R } = { y | y = [ A ] R }
8	5, 6, 7	3eqtr4ri 2655	1 |- { [ A ] R } = ( { A } /. R )

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   {csn 4177   [cec 7740   /.cqs 7741

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-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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  df-qs 7748

This theorem is referenced by: (None)