Theorem dffr2 2909

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 2907	. 2 |- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.w e. x A.v e. x -. vRw))
2		disj 2301	. . . . . . 7 |- ((x i^i {z | zRy}) = (/) <-> A.v e. x -. v e. {z | zRy})
3		visset 1804	. . . . . . . . . 10 |- v e. V
4		breq1 2612	. . . . . . . . . 10 |- (z = v -> (zRy <-> vRy))
5	3, 4	elab 1888	. . . . . . . . 9 |- (v e. {z | zRy} <-> vRy)
6	5	negbii 187	. . . . . . . 8 |- (-. v e. {z | zRy} <-> -. vRy)
7	6	ralbii 1659	. . . . . . 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 1660	. . . . 5 |- (E.y e. x (x i^i {z | zRy}) = (/) <-> E.y e. x A.v e. x -. vRy)
10		breq2 2613	. . . . . . . 8 |- (y = w -> (vRy <-> vRw))
11	10	negbid 609	. . . . . . 7 |- (y = w -> (-. vRy <-> -. vRw))
12	11	ralbidv 1655	. . . . . 6 |- (y = w -> (A.v e. x -. vRy <-> A.v e. x -. vRw))
13	12	cbvrexv 1792	. . . . 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 996	. 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 951   = wceq 953   e. wcel 955  {cab 1456   =/= wne 1577  A.wral 1637  E.wrex 1638   i^i cin 2036   (_ wss 2037  (/)c0 2270   class class class wbr 2609   Fr wfr 2905

This theorem is referenced by:  frc 2910  frss 2911  fr0 2917  dfepfr 2922  dffr3 3415

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-nul 2271  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-fr 2907

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