Theorem dfepfr 2184

Description: A simpler way of saying that the epsilon relation is founded.

Assertion

Ref	Expression
dfepfr	|- (E Fr A <-> A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i y) = (/)))

Distinct variable group(s):   x,y   x,A

Proof of Theorem dfepfr

Step	Hyp	Ref	Expression
1		dffr2 2171	. 2 |- (E Fr A <-> A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i {z | zEy}) = (/)))
2		epel 2124	. . . . . . . . 9 |- (zEy <-> z e. y)
3	2	biabi 1181	. . . . . . . 8 |- {z | zEy} = {z | z e. y}
4		abid2 1186	. . . . . . . 8 |- {z | z e. y} = y
5	3, 4	eqtr 1119	. . . . . . 7 |- {z | zEy} = y
6	5	ineq2i 1642	. . . . . 6 |- (x i^i {z | zEy}) = (x i^i y)
7	6	cleq1i 1108	. . . . 5 |- ((x i^i {z | zEy}) = (/) <-> (x i^i y) = (/))
8	7	birex 1224	. . . 4 |- (E.y e. x (x i^i {z | zEy}) = (/) <-> E.y e. x (x i^i y) = (/))
9	8	imbi2i 160	. . 3 |- (((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i {z | zEy}) = (/)) <-> ((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i y) = (/)))
10	9	bial 695	. 2 |- (A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i {z | zEy}) = (/)) <-> A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i y) = (/)))
11	1, 10	bitr 151	1 |- (E Fr A <-> A.x((x (_ A /\ -. x = (/)) -> E.y e. x (x i^i y) = (/)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wel 803  {cab 1090   = wceq 1091  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707   class class class wbr 2054  Ecep 2056   Fr wfr 2061

This theorem is referenced by:  onfr 2237  zfregfr 3452

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169

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