Theorem dffr4 13893

Description: Alternate definition of founded relation.

Assertion

Ref	Expression
dffr4	|- (R Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x Pred(R, x, y) = (/)))

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

Proof of Theorem dffr4

Step	Hyp	Ref	Expression
1		dffr3 4297	. 2 |- (R Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)))
2		df-pred 13880	. . . . . 6 |- Pred(R, x, y) = (x i^i (`'R"{y}))
3	2	eqeq1i 1891	. . . . 5 |- (Pred(R, x, y) = (/) <-> (x i^i (`'R"{y})) = (/))
4	3	rexbii 2128	. . . 4 |- (E.y e. x Pred(R, x, y) = (/) <-> E.y e. x (x i^i (`'R"{y})) = (/))
5	4	imbi2i 202	. . 3 |- (((x C_ A /\ x =/= (/)) -> E.y e. x Pred(R, x, y) = (/)) <-> ((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)))
6	5	albii 1346	. 2 |- (A.x((x C_ A /\ x =/= (/)) -> E.y e. x Pred(R, x, y) = (/)) <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)))
7	1, 6	bitr4i 193	1 |- (R Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x Pred(R, x, y) = (/)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044   Fr wfr 3623  `'ccnv 3985  "cima 3989  Predcpred 13879

This theorem is referenced by:  tz6.26 13913  frmin 13938

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-fr 3625  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-pred 13880

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