Theorem dffr3 3431

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 2919	. 2 |- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		visset 1813	. . . . . . . 8 |- y e. V
3		iniseg 3430	. . . . . . . 8 |- (y e. V -> (`'R"{y}) = {z | zRy})
4	2, 3	ax-mp 7	. . . . . . 7 |- (`'R"{y}) = {z | zRy}
5	4	ineq2i 2214	. . . . . 6 |- (x i^i (`'R"{y})) = (x i^i {z | zRy})
6	5	eqeq1i 1482	. . . . 5 |- ((x i^i (`'R"{y})) = (/) <-> (x i^i {z | zRy}) = (/))
7	6	rexbii 1668	. . . 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 999	. 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 954   = wceq 956   e. wcel 958  {cab 1463   =/= wne 1585  E.wrex 1646  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409   class class class wbr 2619   Fr wfr 2915  `'ccnv 3169  "cima 3173

This theorem is referenced by:  isofrlem 3901

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-fr 2917  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

Copyright terms: Public domain