Theorem dffr5 31643

Description: A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)

Assertion

Ref	Expression
dffr5	|- ( R Fr A <-> ( ~P A \ { (/) } ) C_ ran ( _E \ ( _E o. `' R ) ) )

Proof of Theorem dffr5

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

Step	Hyp	Ref	Expression
1		eldif 3584	. . . . 5 |- ( x e. ( ~P A \ { (/) } ) <-> ( x e. ~P A /\ -. x e. { (/) } ) )
2		selpw 4165	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
3		velsn 4193	. . . . . . 7 |- ( x e. { (/) } <-> x = (/) )
4	3	necon3bbii 2841	. . . . . 6 |- ( -. x e. { (/) } <-> x =/= (/) )
5	2, 4	anbi12i 733	. . . . 5 |- ( ( x e. ~P A /\ -. x e. { (/) } ) <-> ( x C_ A /\ x =/= (/) ) )
6	1, 5	bitri 264	. . . 4 |- ( x e. ( ~P A \ { (/) } ) <-> ( x C_ A /\ x =/= (/) ) )
7		brdif 4705	. . . . . . 7 |- ( y ( _E \ ( _E o. `' R ) ) x <-> ( y _E x /\ -. y ( _E o. `' R ) x ) )
8		epel 5032	. . . . . . . 8 |- ( y _E x <-> y e. x )
9		vex 3203	. . . . . . . . . . 11 |- y e. _V
10		vex 3203	. . . . . . . . . . 11 |- x e. _V
11	9, 10	coep 31641	. . . . . . . . . 10 |- ( y ( _E o. `' R ) x <-> E. z e. x y `' R z )
12		vex 3203	. . . . . . . . . . . 12 |- z e. _V
13	9, 12	brcnv 5305	. . . . . . . . . . 11 |- ( y `' R z <-> z R y )
14	13	rexbii 3041	. . . . . . . . . 10 |- ( E. z e. x y `' R z <-> E. z e. x z R y )
15		dfrex2 2996	. . . . . . . . . 10 |- ( E. z e. x z R y <-> -. A. z e. x -. z R y )
16	11, 14, 15	3bitrri 287	. . . . . . . . 9 |- ( -. A. z e. x -. z R y <-> y ( _E o. `' R ) x )
17	16	con1bii 346	. . . . . . . 8 |- ( -. y ( _E o. `' R ) x <-> A. z e. x -. z R y )
18	8, 17	anbi12i 733	. . . . . . 7 |- ( ( y _E x /\ -. y ( _E o. `' R ) x ) <-> ( y e. x /\ A. z e. x -. z R y ) )
19	7, 18	bitri 264	. . . . . 6 |- ( y ( _E \ ( _E o. `' R ) ) x <-> ( y e. x /\ A. z e. x -. z R y ) )
20	19	exbii 1774	. . . . 5 |- ( E. y y ( _E \ ( _E o. `' R ) ) x <-> E. y ( y e. x /\ A. z e. x -. z R y ) )
21	10	elrn 5366	. . . . 5 |- ( x e. ran ( _E \ ( _E o. `' R ) ) <-> E. y y ( _E \ ( _E o. `' R ) ) x )
22		df-rex 2918	. . . . 5 |- ( E. y e. x A. z e. x -. z R y <-> E. y ( y e. x /\ A. z e. x -. z R y ) )
23	20, 21, 22	3bitr4i 292	. . . 4 |- ( x e. ran ( _E \ ( _E o. `' R ) ) <-> E. y e. x A. z e. x -. z R y )
24	6, 23	imbi12i 340	. . 3 |- ( ( x e. ( ~P A \ { (/) } ) -> x e. ran ( _E \ ( _E o. `' R ) ) ) <-> ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
25	24	albii 1747	. 2 |- ( A. x ( x e. ( ~P A \ { (/) } ) -> x e. ran ( _E \ ( _E o. `' R ) ) ) <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
26		dfss2 3591	. 2 |- ( ( ~P A \ { (/) } ) C_ ran ( _E \ ( _E o. `' R ) ) <-> A. x ( x e. ( ~P A \ { (/) } ) -> x e. ran ( _E \ ( _E o. `' R ) ) ) )
27		df-fr 5073	. 2 |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
28	25, 26, 27	3bitr4ri 293	1 |- ( R Fr A <-> ( ~P A \ { (/) } ) C_ ran ( _E \ ( _E o. `' R ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   _E cep 5028   Fr wfr 5070   `'ccnv 5113   ran crn 5115   o. ccom 5118

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)