Theorem dffr3 3437

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

Assertion

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

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

Proof of Theorem dffr3

Step	Hyp	Ref	Expression
1		dffr2 2925	. 2 |- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		visset 1816	. . . . . . . 8 |- y e. V
3		iniseg 3436	. . . . . . . 8 |- (y e. V -> (`'R"{y}) = {z | zRy})
4	2, 3	ax-mp 7	. . . . . . 7 |- (`'R"{y}) = {z | zRy}
5	4	ineq2i 2217	. . . . . 6 |- (x i^i (`'R"{y})) = (x i^i {z | zRy})
6	5	eqeq1i 1485	. . . . 5 |- ((x i^i (`'R"{y})) = (/) <-> (x i^i {z | zRy}) = (/))
7	6	rexbii 1671	. . . 4 |- (E.y e. x (x i^i (`'R"{y})) = (/) <-> E.y e. x (x i^i {z | zRy}) = (/))
8	7	imbi2i 185	. . 3 |- (((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)) <-> ((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
9	8	albii 1001	. 2 |- (A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)) <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
10	1, 9	bitr4 176	1 |- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  {cab 1466   =/= wne 1588  E.wrex 1649  Vcvv 1814   i^i cin 2049   (_ wss 2050  (/)c0 2283  {csn 2413   class class class wbr 2624   Fr wfr 2921  `'ccnv 3175  "cima 3179

This theorem is referenced by:  isofrlem 3907

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-11 969  ax-12 970  ax-13 971  ax-14 972  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  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-fr 2923  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197

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