Theorem dffr3 5498

Description: Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)

Assertion

Ref	Expression
dffr3	|- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) )

Distinct variable groups:   x, y, A   x, R, y

Proof of Theorem dffr3

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffr2 5079	. 2 |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
2		vex 3203	. . . . . . . . 9 |- y e. _V
3		iniseg 5496	. . . . . . . . 9 |- ( y e. _V -> ( `' R " { y } ) = { z | z R y } )
4	2, 3	ax-mp 5	. . . . . . . 8 |- ( `' R " { y } ) = { z | z R y }
5	4	ineq2i 3811	. . . . . . 7 |- ( x i^i ( `' R " { y } ) ) = ( x i^i { z | z R y } )
6		dfrab3 3902	. . . . . . 7 |- { z e. x | z R y } = ( x i^i { z | z R y } )
7	5, 6	eqtr4i 2647	. . . . . 6 |- ( x i^i ( `' R " { y } ) ) = { z e. x | z R y }
8	7	eqeq1i 2627	. . . . 5 |- ( ( x i^i ( `' R " { y } ) ) = (/) <-> { z e. x | z R y } = (/) )
9	8	rexbii 3041	. . . 4 |- ( E. y e. x ( x i^i ( `' R " { y } ) ) = (/) <-> E. y e. x { z e. x | z R y } = (/) )
10	9	imbi2i 326	. . 3 |- ( ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) <-> ( ( x C_ A /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
11	10	albii 1747	. 2 |- ( A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
12	1, 11	bitr4i 267	1 |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   Fr wfr 5070   `'ccnv 5113   "cima 5117

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-ral 2917  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-fr 5073  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  dffr4  5696  isofrlem  6590