Theorem dffr2 2171

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 group(s):   x,y,z,R   x,A

Proof of Theorem dffr2

Step	Hyp	Ref	Expression
1		df-fr 2169	. 2 |- (R Fr A <-> A.x((x (_ A /\ -. x = (/)) -> E.v e. x A.w e. x -. wRv))
2		disj 1733	. . . . . . 7 |- ((x i^i {z | zRy}) = (/) <-> A.w e. x -. w e. {z | zRy})
3		visset 1350	. . . . . . . . . 10 |- w e. V
4		breq1 2065	. . . . . . . . . 10 |- (z = w -> (zRy <-> wRy))
5	3, 4	elab 1415	. . . . . . . . 9 |- (w e. {z | zRy} <-> wRy)
6	5	negbii 162	. . . . . . . 8 |- (-. w e. {z | zRy} <-> -. wRy)
7	6	biral 1223	. . . . . . 7 |- (A.w e. x -. w e. {z | zRy} <-> A.w e. x -. wRy)
8	2, 7	bitr 151	. . . . . 6 |- ((x i^i {z | zRy}) = (/) <-> A.w e. x -. wRy)
9	8	birex 1224	. . . . 5 |- (E.y e. x (x i^i {z | zRy}) = (/) <-> E.y e. x A.w e. x -. wRy)
10		breq2 2066	. . . . . . . 8 |- (y = v -> (wRy <-> wRv))
11	10	negbid 463	. . . . . . 7 |- (y = v -> (-. wRy <-> -. wRv))
12	11	biraldv 1219	. . . . . 6 |- (y = v -> (A.w e. x -. wRy <-> A.w e. x -. wRv))
13	12	cbvrexv 1334	. . . . 5 |- (E.y e. x A.w e. x -. wRy <-> E.v e. x A.w e. x -. wRv)
14	9, 13	bitr 151	. . . 4 |- (E.y e. x (x i^i {z | zRy}) = (/) <-> E.v e. x A.w e. x -. wRv)
15	14	imbi2i 160	. . 3 |- (((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i {z | zRy}) = (/)) <-> ((x (_ A /\ -. x = (/)) -> E.v e. x A.w e. x -. wRv))
16	15	bial 695	. 2 |- (A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i {z | zRy}) = (/)) <-> A.x((x (_ A /\ -. x = (/)) -> E.v e. x A.w e. x -. wRv))
17	1, 16	bitr4 154	1 |- (R Fr A <-> A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  frc 2172  frss 2173  fr0 2179  dfepfr 2184  dffr3 2620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-fr 2169

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