Definition df-nninf 7121

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 9242 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 6433) 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 7120	. 2 class ℕ∞
2		vi	. . . . . . . 8 setvar i
3	2	cv 1352	. . . . . . 7 class i
4	3	csuc 4367	. . . . . 6 class suc i
5		vf	. . . . . . 7 setvar f
6	5	cv 1352	. . . . . 6 class f
7	4, 6	cfv 5218	. . . . 5 class ( f ` suc i )
8	3, 6	cfv 5218	. . . . 5 class ( f ` i )
9	7, 8	wss 3131	. . . 4 wff ( f ` suc i ) C_ ( f ` i )
10		com 4591	. . . 4 class om
11	9, 2, 10	wral 2455	. . 3 wff A. i e. om ( f ` suc i ) C_ ( f ` i )
12		c2o 6413	. . . 4 class 2o
13		cmap 6650	. . . 4 class ^m
14	12, 10, 13	co 5877	. . 3 class ( 2o ^m om )
15	11, 5, 14	crab 2459	. 2 class { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }
16	1, 15	wceq 1353	1 wff ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }

This definition is referenced by:  nninfex  7122  nninff  7123  infnninf  7124  infnninfOLD  7125  nnnninf  7126  nnnninfeq  7128  nnnninfeq2  7129  nninfwlpoimlemg  7175  0nninf  14792  nnsf  14793  peano4nninf  14794  nninfalllem1  14796  nninfself  14801