Definition df-nninf 7461

Description: Define the set of nonincreasing sequences in 2o ^m om. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as NN0* as defined at df-xnn0 9636 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used om or NN0, but the former allows us to take advantage of 2o = { (/) , 1o } (df2o3 6702) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)

Assertion

Ref	Expression
df-nninf	|- ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }

Distinct variable group:   f, i

Detailed syntax breakdown of Definition df-nninf

Step	Hyp	Ref	Expression
1		xnninf 7460	. 2 class ℕ∞
2		vi	. . . . . . . 8 setvar i
3	2	cv 1401	. . . . . . 7 class i
4	3	csuc 4510	. . . . . 6 class suc i
5		vf	. . . . . . 7 setvar f
6	5	cv 1401	. . . . . 6 class f
7	4, 6	cfv 5377	. . . . 5 class ( f ` suc i )
8	3, 6	cfv 5377	. . . . 5 class ( f ` i )
9	7, 8	wss 3220	. . . 4 wff ( f ` suc i ) C_ ( f ` i )
10		com 4737	. . . 4 class om
11	9, 2, 10	wral 2528	. . 3 wff A. i e. om ( f ` suc i ) C_ ( f ` i )
12		c2o 6681	. . . 4 class 2o
13		cmap 6922	. . . 4 class ^m
14	12, 10, 13	co 6085	. . 3 class ( 2o ^m om )
15	11, 5, 14	crab 2532	. 2 class { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }
16	1, 15	wceq 1402	1 wff ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }

This definition is used by:  nninfex  7462  nninff  7463  nninfninc  7464  infnninf  7465  infnninfOLD  7466  nnnninf  7467  nnnninfeq  7469  nnnninfeq2  7470  nninfwlpoimlemg  7516  0nninf  17213  nnsf  17214  peano4nninf  17215  nninfalllem1  17217  nninfself  17222