Theorem dffr2 2925

Description: Alternate definition of founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
dffr2	|- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))

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

Proof of Theorem dffr2

Step	Hyp	Ref	Expression
1		df-fr 2923	. 2 |- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.w e. x A.v e. x -. vRw))
2		disj 2315	. . . . . . 7 |- ((x i^i {z | zRy}) = (/) <-> A.v e. x -. v e. {z | zRy})
3		visset 1816	. . . . . . . . . 10 |- v e. V
4		breq1 2627	. . . . . . . . . 10 |- (z = v -> (zRy <-> vRy))
5	3, 4	elab 1900	. . . . . . . . 9 |- (v e. {z | zRy} <-> vRy)
6	5	negbii 187	. . . . . . . 8 |- (-. v e. {z | zRy} <-> -. vRy)
7	6	ralbii 1670	. . . . . . 7 |- (A.v e. x -. v e. {z | zRy} <-> A.v e. x -. vRy)
8	2, 7	bitr 173	. . . . . 6 |- ((x i^i {z | zRy}) = (/) <-> A.v e. x -. vRy)
9	8	rexbii 1671	. . . . 5 |- (E.y e. x (x i^i {z | zRy}) = (/) <-> E.y e. x A.v e. x -. vRy)
10		breq2 2628	. . . . . . . 8 |- (y = w -> (vRy <-> vRw))
11	10	negbid 613	. . . . . . 7 |- (y = w -> (-. vRy <-> -. vRw))
12	11	ralbidv 1666	. . . . . 6 |- (y = w -> (A.v e. x -. vRy <-> A.v e. x -. vRw))
13	12	cbvrexv 1804	. . . . 5 |- (E.y e. x A.v e. x -. vRy <-> E.w e. x A.v e. x -. vRw)
14	9, 13	bitr 173	. . . 4 |- (E.y e. x (x i^i {z | zRy}) = (/) <-> E.w e. x A.v e. x -. vRw)
15	14	imbi2i 185	. . 3 |- (((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)) <-> ((x (_ A /\ x =/= (/)) -> E.w e. x A.v e. x -. vRw))
16	15	albii 1001	. 2 |- (A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)) <-> A.x((x (_ A /\ x =/= (/)) -> E.w e. x A.v e. x -. vRw))
17	1, 16	bitr4 176	1 |- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  {cab 1466   =/= wne 1588  A.wral 1648  E.wrex 1649   i^i cin 2049   (_ wss 2050  (/)c0 2283   class class class wbr 2624   Fr wfr 2921

This theorem is referenced by:  frc 2926  frss 2927  fr0 2933  dfepfr 2938  dffr3 3437

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-fr 2923

Copyright terms: Public domain