Definition df-nninf 7298

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 9444 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 6583) 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 7297	. 2 class ℕ∞
2		vi	. . . . . . . 8 setvar i
3	2	cv 1394	. . . . . . 7 class i
4	3	csuc 4456	. . . . . 6 class suc i
5		vf	. . . . . . 7 setvar f
6	5	cv 1394	. . . . . 6 class f
7	4, 6	cfv 5318	. . . . 5 class ( f ` suc i )
8	3, 6	cfv 5318	. . . . 5 class ( f ` i )
9	7, 8	wss 3197	. . . 4 wff ( f ` suc i ) C_ ( f ` i )
10		com 4682	. . . 4 class om
11	9, 2, 10	wral 2508	. . 3 wff A. i e. om ( f ` suc i ) C_ ( f ` i )
12		c2o 6562	. . . 4 class 2o
13		cmap 6803	. . . 4 class ^m
14	12, 10, 13	co 6007	. . 3 class ( 2o ^m om )
15	11, 5, 14	crab 2512	. 2 class { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }
16	1, 15	wceq 1395	1 wff ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }

This definition is referenced by:  nninfex  7299  nninff  7300  nninfninc  7301  infnninf  7302  infnninfOLD  7303  nnnninf  7304  nnnninfeq  7306  nnnninfeq2  7307  nninfwlpoimlemg  7353  0nninf  16430  nnsf  16431  peano4nninf  16432  nninfalllem1  16434  nninfself  16439