Definition df-fr 2907

Description: Define a founded relation. For alternate definitions, see dffr2 2909 and dffr3 3415.

Assertion

Ref	Expression
df-fr	|- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x A.z e. x -. zRy))

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

Detailed syntax breakdown of Definition df-fr

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cR	. . 3 class R
3	1, 2	wfr 2905	. 2 wff R Fr A
4		vx	. . . . . . 7 set x
5	4	cv 952	. . . . . 6 class x
6	5, 1	wss 2037	. . . . 5 wff x (_ A
7		c0 2270	. . . . . 6 class (/)
8	5, 7	wne 1577	. . . . 5 wff x =/= (/)
9	6, 8	wa 223	. . . 4 wff (x (_ A /\ x =/= (/))
10		vz	. . . . . . . . 9 set z
11	10	cv 952	. . . . . . . 8 class z
12		vy	. . . . . . . . 9 set y
13	12	cv 952	. . . . . . . 8 class y
14	11, 13, 2	wbr 2609	. . . . . . 7 wff zRy
15	14	wn 2	. . . . . 6 wff -. zRy
16	15, 10, 5	wral 1637	. . . . 5 wff A.z e. x -. zRy
17	16, 12, 5	wrex 1638	. . . 4 wff E.y e. x A.z e. x -. zRy
18	9, 17	wi 3	. . 3 wff ((x (_ A /\ x =/= (/)) -> E.y e. x A.z e. x -. zRy)
19	18, 4	wal 951	. 2 wff A.x((x (_ A /\ x =/= (/)) -> E.y e. x A.z e. x -. zRy)
20	3, 19	wb 146	1 wff (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x A.z e. x -. zRy))

This definition is referenced by:  fri 2908  dffr2 2909  freq1 2912  weinxp 3223  f1oweALT 3891

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