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Definition df-nninf 7000
Description: Define the set of nonincreasing sequences in 2o𝑚 ω. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as 0* as defined at df-xnn0 9034 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used ω or 0, but the former allows us to take advantage of 2o = {∅, 1o} (df2o3 6320) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
Assertion
Ref Expression
df-nninf = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
Distinct variable group:   𝑓,𝑖

Detailed syntax breakdown of Definition df-nninf
StepHypRef Expression
1 xnninf 6998 . 2 class
2 vi . . . . . . . 8 setvar 𝑖
32cv 1330 . . . . . . 7 class 𝑖
43csuc 4282 . . . . . 6 class suc 𝑖
5 vf . . . . . . 7 setvar 𝑓
65cv 1330 . . . . . 6 class 𝑓
74, 6cfv 5118 . . . . 5 class (𝑓‘suc 𝑖)
83, 6cfv 5118 . . . . 5 class (𝑓𝑖)
97, 8wss 3066 . . . 4 wff (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)
10 com 4499 . . . 4 class ω
119, 2, 10wral 2414 . . 3 wff 𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)
12 c2o 6300 . . . 4 class 2o
13 cmap 6535 . . . 4 class 𝑚
1412, 10, 13co 5767 . . 3 class (2o𝑚 ω)
1511, 5, 14crab 2418 . 2 class {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
161, 15wceq 1331 1 wff = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
Colors of variables: wff set class
This definition is referenced by:  infnninf  7015  nnnninf  7016  0nninf  13186  nninff  13187  nnsf  13188  peano4nninf  13189  nninfalllemn  13191  nninfalllem1  13192  nninfex  13194  nninfself  13198
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