Definition df-cap 31977

Description: Define the little cap function. See brcap 32047 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)

Assertion

Ref	Expression
df-cap	|- Cap = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V ) ) )

Detailed syntax breakdown of Definition df-cap

Step	Hyp	Ref	Expression
1		ccap 31954	. 2 class Cap
2		cvv 3200	. . . . 5 class _V
3	2, 2	cxp 5112	. . . 4 class ( _V X. _V )
4	3, 2	cxp 5112	. . 3 class ( ( _V X. _V ) X. _V )
5		cep 5028	. . . . . 6 class _E
6	2, 5	ctxp 31937	. . . . 5 class ( _V (x) _E )
7		c1st 7166	. . . . . . . . 9 class 1st
8	7	ccnv 5113	. . . . . . . 8 class `' 1st
9	8, 5	ccom 5118	. . . . . . 7 class ( `' 1st o. _E )
10		c2nd 7167	. . . . . . . . 9 class 2nd
11	10	ccnv 5113	. . . . . . . 8 class `' 2nd
12	11, 5	ccom 5118	. . . . . . 7 class ( `' 2nd o. _E )
13	9, 12	cin 3573	. . . . . 6 class ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) )
14	13, 2	ctxp 31937	. . . . 5 class ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V )
15	6, 14	csymdif 3843	. . . 4 class ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V ) )
16	15	crn 5115	. . 3 class ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V ) )
17	4, 16	cdif 3571	. 2 class ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V ) ) )
18	1, 17	wceq 1483	1 wff Cap = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) i^i ( `' 2nd o. _E ) ) (x) _V ) ) )

This definition is referenced by:  brcap  32047