Definition df-ihash 11193

Description: Define the set size function ♯, which gives the cardinality of a finite set as a member of NN0, and assigns all infinite sets the value +oo. For example, (♯ ` { 0 , 1 , 2 } ) = 3.

Since we don't know that an arbitrary set is either finite or infinite (by inffiexmid 7203), the behavior beyond finite sets is not as useful as it might appear. For example, we wouldn't expect to be able to define this function in a meaningful way on ~P 1o, which cannot be shown to be finite (per pw1fin 7207).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8900). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of U. and ~<_) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

Assertion

Ref	Expression
df-ihash	|- ♯ = ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } ) o. ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-ihash

Step	Hyp	Ref	Expression
1		chash 11192	. 2 class ♯
2		vx	. . . . . 6 setvar x
3		cz 9623	. . . . . 6 class ZZ
4	2	cv 1401	. . . . . . 7 class x
5		c1 8170	. . . . . . 7 class 1
6		caddc 8172	. . . . . . 7 class +
7	4, 5, 6	co 6075	. . . . . 6 class ( x + 1 )
8	2, 3, 7	cmpt 4187	. . . . 5 class ( x e. ZZ |-> ( x + 1 ) )
9		cc0 8169	. . . . 5 class 0
10	8, 9	cfrec 6651	. . . 4 class frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11		com 4732	. . . . . 6 class om
12		cpnf 8347	. . . . . 6 class +oo
13	11, 12	cop 3708	. . . . 5 class <. om , +oo >.
14	13	csn 3705	. . . 4 class { <. om , +oo >. }
15	10, 14	cun 3218	. . 3 class (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } )
16		cvv 2821	. . . 4 class _V
17		vy	. . . . . . . 8 setvar y
18	17	cv 1401	. . . . . . 7 class y
19		cdom 7011	. . . . . . 7 class ~<_
20	18, 4, 19	wbr 4125	. . . . . 6 wff y ~<_ x
21	11	csn 3705	. . . . . . 7 class { om }
22	11, 21	cun 3218	. . . . . 6 class ( om u. { om } )
23	20, 17, 22	crab 2532	. . . . 5 class { y e. ( om u. { om } ) | y ~<_ x }
24	23	cuni 3930	. . . 4 class U. { y e. ( om u. { om } ) | y ~<_ x }
25	2, 16, 24	cmpt 4187	. . 3 class ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } )
26	15, 25	ccom 4773	. 2 class ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } ) o. ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } ) )
27	1, 26	wceq 1402	1 wff ♯ = ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } ) o. ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } ) )

This definition is referenced by:  hashinfom  11195  hashennn  11197