Theorem frsn 5059

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 4827	. . . 4 |- ( R Fr { A } <-> A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
2		df-ne 2651	. . . . . . . . . 10 |- ( x =/= (/) <-> -. x = (/) )
3		simpr 459	. . . . . . . . . . . 12 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> x C_ { A } )
4		sssn 4174	. . . . . . . . . . . 12 |- ( x C_ { A } <-> ( x = (/) \/ x = { A } ) )
5	3, 4	sylib 196	. . . . . . . . . . 11 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> ( x = (/) \/ x = { A } ) )
6	5	ord 375	. . . . . . . . . 10 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> ( -. x = (/) -> x = { A } ) )
7	2, 6	syl5bi 217	. . . . . . . . 9 |- ( ( ( Rel R /\ A e. _V ) /\ x C_ { A } ) -> ( x =/= (/) -> x = { A } ) )
8	7	impr 617	. . . . . . . 8 |- ( ( ( Rel R /\ A e. _V ) /\ ( x C_ { A } /\ x =/= (/) ) ) -> x = { A } )
9		eqimss 3541	. . . . . . . . . 10 |- ( x = { A } -> x C_ { A } )
10	9	adantl 464	. . . . . . . . 9 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> x C_ { A } )
11		simpr 459	. . . . . . . . . 10 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> x = { A } )
12		snnzg 4133	. . . . . . . . . . 11 |- ( A e. _V -> { A } =/= (/) )
13	12	ad2antlr 724	. . . . . . . . . 10 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> { A } =/= (/) )
14	11, 13	eqnetrd 2747	. . . . . . . . 9 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> x =/= (/) )
15	10, 14	jca 530	. . . . . . . 8 |- ( ( ( Rel R /\ A e. _V ) /\ x = { A } ) -> ( x C_ { A } /\ x =/= (/) ) )
16	8, 15	impbida 830	. . . . . . 7 |- ( ( Rel R /\ A e. _V ) -> ( ( x C_ { A } /\ x =/= (/) ) <-> x = { A } ) )
17	16	imbi1d 315	. . . . . 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 1718	. . . . 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 4678	. . . . . 6 |- { A } e. _V
20		raleq 3051	. . . . . . 7 |- ( x = { A } -> ( A. z e. x -. z R y <-> A. z e. { A } -. z R y ) )
21	20	rexeqbi1dv 3060	. . . . . 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 3134	. . . . 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 261	. . . 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 257	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( R Fr { A } <-> E. y e. { A } A. z e. { A } -. z R y ) )
25		breq2 4443	. . . . . . . 8 |- ( y = A -> ( z R y <-> z R A ) )
26	25	notbid 292	. . . . . . 7 |- ( y = A -> ( -. z R y <-> -. z R A ) )
27	26	ralbidv 2893	. . . . . 6 |- ( y = A -> ( A. z e. { A } -. z R y <-> A. z e. { A } -. z R A ) )
28	27	rexsng 4052	. . . . 5 |- ( A e. _V -> ( E. y e. { A } A. z e. { A } -. z R y <-> A. z e. { A } -. z R A ) )
29		breq1 4442	. . . . . . 7 |- ( z = A -> ( z R A <-> A R A ) )
30	29	notbid 292	. . . . . 6 |- ( z = A -> ( -. z R A <-> -. A R A ) )
31	30	ralsng 4051	. . . . 5 |- ( A e. _V -> ( A. z e. { A } -. z R A <-> -. A R A ) )
32	28, 31	bitrd 253	. . . 4 |- ( A e. _V -> ( E. y e. { A } A. z e. { A } -. z R y <-> -. A R A ) )
33	32	adantl 464	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( E. y e. { A } A. z e. { A } -. z R y <-> -. A R A ) )
34	24, 33	bitrd 253	. 2 |- ( ( Rel R /\ A e. _V ) -> ( R Fr { A } <-> -. A R A ) )
35		snprc 4079	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
36		fr0 4847	. . . . . 6 |- R Fr (/)
37		freq2 4839	. . . . . 6 |- ( { A } = (/) -> ( R Fr { A } <-> R Fr (/) ) )
38	36, 37	mpbiri 233	. . . . 5 |- ( { A } = (/) -> R Fr { A } )
39	35, 38	sylbi 195	. . . 4 |- ( -. A e. _V -> R Fr { A } )
40	39	adantl 464	. . 3 |- ( ( Rel R /\ -. A e. _V ) -> R Fr { A } )
41		brrelex 5027	. . . 4 |- ( ( Rel R /\ A R A ) -> A e. _V )
42	41	stoic1a 1609	. . 3 |- ( ( Rel R /\ -. A e. _V ) -> -. A R A )
43	40, 42	2thd 240	. 2 |- ( ( Rel R /\ -. A e. _V ) -> ( R Fr { A } <-> -. A R A ) )
44	34, 43	pm2.61dan 789	1 |- ( Rel R -> ( R Fr { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   A.wal 1396   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439   Fr wfr 4824   Rel wrel 4993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-fr 4827  df-xp 4994  df-rel 4995

This theorem is referenced by:  wesn  5060