Theorem frsn 4882

Description: Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
frsn	|- ( Rel R -> ( R Fr { A } <-> -. A R A ) )

Proof of Theorem frsn

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

Step	Hyp	Ref	Expression
1		df-fr 4770	. . . 4 |- ( R Fr { A } <-> A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
2		df-ne 2623	. . . . . . . . . 10 |- ( x =/= (/) <-> -. x = (/) )
3		simpr 467	. . . . . . . . . . . 12 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> x C_ { A } )
4		sssn 4098	. . . . . . . . . . . 12 |- ( x C_ { A } <-> ( x = (/) \/ x = { A } ) )
5	3, 4	sylib 201	. . . . . . . . . . 11 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> ( x = (/) \/ x = { A } ) )
6	5	ord 383	. . . . . . . . . 10 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> ( -. x = (/) -> x = { A } ) )
7	2, 6	syl5bi 225	. . . . . . . . 9 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> ( x =/= (/) -> x = { A } ) )
8	7	impr 629	. . . . . . . 8 |- ( ( ( Rel R /\ A e. _V ) /\ ( x C_ { A } /\ x =/= (/) ) ) -> x = { A } )
9		eqimss 3451	. . . . . . . . . 10 |- ( x = { A } -> x C_ { A } )
10	9	adantl 472	. . . . . . . . 9 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> x C_ { A } )
11		simpr 467	. . . . . . . . . 10 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> x = { A } )
12		snnzg 4057	. . . . . . . . . . 11 |- ( A e. _V -> { A } =/= (/) )
13	12	ad2antlr 738	. . . . . . . . . 10 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> { A } =/= (/) )
14	11, 13	eqnetrd 2690	. . . . . . . . 9 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> x =/= (/) )
15	10, 14	jca 539	. . . . . . . 8 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> ( x C_ { A } /\ x =/= (/) ) )
16	8, 15	impbida 847	. . . . . . 7 |- ( ( Rel R /\ A e. _V ) -> ( ( x C_ { A } /\ x =/= (/) ) <-> x = { A } ) )
17	16	imbi1d 323	. . . . . 6 |- ( ( Rel R /\ A e. _V ) -> ( ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> ( x = { A } -> E. y e. x A. z e. x -. z R y ) ) )
18	17	albidv 1770	. . . . 5 |- ( ( Rel R /\ A e. _V ) -> ( A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> A. x ( x = { A } -> E. y e. x A. z e. x -. z R y ) ) )
19		snex 4613	. . . . . 6 |- { A } e. _V
20		raleq 2954	. . . . . . 7 |- ( x = { A } -> ( A. z e. x -. z R y <-> A. z e. { A } -. z R y ) )
21	20	rexeqbi1dv 2963	. . . . . 6 |- ( x = { A } -> ( E. y e. x A. z e. x -. z R y <-> E. y e. { A } A. z e. { A } -. z R y ) )
22	19, 21	ceqsalv 3042	. . . . 5 |- ( A. x ( x = { A } -> E. y e. x A. z e. x -. z R y ) <-> E. y e. { A } A. z e. { A } -. z R y )
23	18, 22	syl6bb 269	. . . 4 |- ( ( Rel R /\ A e. _V ) -> ( A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> E. y e. { A } A. z e. { A } -. z R y ) )
24	1, 23	syl5bb 265	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( R Fr { A } <-> E. y e. { A } A. z e. { A } -. z R y ) )
25		breq2 4377	. . . . . . . 8 |- ( y = A -> ( z R y <-> z R A ) )
26	25	notbid 300	. . . . . . 7 |- ( y = A -> ( -. z R y <-> -. z R A ) )
27	26	ralbidv 2809	. . . . . 6 |- ( y = A -> ( A. z e. { A } -. z R y <-> A. z e. { A } -. z R A ) )
28	27	rexsng 3974	. . . . 5 |- ( A e. _V -> ( E. y e. { A } A. z e. { A } -. z R y <-> A. z e. { A } -. z R A ) )
29		breq1 4376	. . . . . . 7 |- ( z = A -> ( z R A <-> A R A ) )
30	29	notbid 300	. . . . . 6 |- ( z = A -> ( -. z R A <-> -. A R A ) )
31	30	ralsng 3973	. . . . 5 |- ( A e. _V -> ( A. z e. { A } -. z R A <-> -. A R A ) )
32	28, 31	bitrd 261	. . . 4 |- ( A e. _V -> ( E. y e. { A } A. z e. { A } -. z R y <-> -. A R A ) )
33	32	adantl 472	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( E. y e. { A } A. z e. { A } -. z R y <-> -. A R A ) )
34	24, 33	bitrd 261	. 2 |- ( ( Rel R /\ A e. _V ) -> ( R Fr { A } <-> -. A R A ) )
35		snprc 4003	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
36		fr0 4790	. . . . . 6 |- R Fr (/)
37		freq2 4782	. . . . . 6 |- ( { A } = (/) -> ( R Fr { A } <-> R Fr (/) ) )
38	36, 37	mpbiri 241	. . . . 5 |- ( { A } = (/) -> R Fr { A } )
39	35, 38	sylbi 200	. . . 4 |- ( -. A e. _V -> R Fr { A } )
40	39	adantl 472	. . 3 |- ( ( Rel R /\ -. A e. _V ) -> R Fr { A } )
41		brrelex 4850	. . . 4 |- ( ( Rel R /\ A R A ) -> A e. _V )
42	41	stoic1a 1658	. . 3 |- ( ( Rel R /\ -. A e. _V ) -> -. A R A )
43	40, 42	2thd 248	. 2 |- ( ( Rel R /\ -. A e. _V ) -> ( R Fr { A } <-> -. A R A ) )
44	34, 43	pm2.61dan 805	1 |- ( Rel R -> ( R Fr { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   A.wal 1445   = wceq 1447   e. wcel 1890   =/= wne 2621   A.wral 2736   E.wrex 2737   _Vcvv 3012   C_ wss 3371   (/)c0 3698   {csn 3935   class class class wbr 4373   Fr wfr 4767   Rel wrel 4816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-br 4374  df-opab 4433  df-fr 4770  df-xp 4817  df-rel 4818

This theorem is referenced by:  wesn  4883