Theorem nninfctlemfo 12411

Description: Lemma for nninfct 12412. (Contributed by Jim Kingdon, 10-Jul-2025.)

Hypotheses

Ref	Expression
nninfct.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
nninfct.f	⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
nninfct.i	⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})

Assertion

Ref	Expression
nninfctlemfo	⊢ (ω ∈ Omni → 𝐼:ℕ0*–onto→ℕ∞)

Distinct variable group:   𝑖,𝐺,𝑛

Allowed substitution hints:   𝐹(𝑥,𝑖,𝑛)   𝐺(𝑥)   𝐼(𝑥,𝑖,𝑛)

Proof of Theorem nninfctlemfo

Dummy variables 𝑘 𝑚 𝑧 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfct.g	. . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
2		nninfct.f	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
3		nninfct.i	. . . 4 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
4	1, 2, 3	fxnn0nninf 10597	. . 3 ⊢ 𝐼:ℕ0*⟶ℕ∞
5	4	a1i 9	. 2 ⊢ (ω ∈ Omni → 𝐼:ℕ0*⟶ℕ∞)
6		ssrab2 3280	. . . . . . . . 9 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} ⊆ (ℤ≥‘0)
7		nn0uz 9696	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
8		nn0ssxnn0 9374	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℕ0*
9	7, 8	eqsstrri 3228	. . . . . . . . 9 ⊢ (ℤ≥‘0) ⊆ ℕ0*
10	6, 9	sstri 3204	. . . . . . . 8 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} ⊆ ℕ0*
11		0zd 9397	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 0 ∈ ℤ)
12		eqid 2206	. . . . . . . . 9 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} = {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}
13		fveq2 5586	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑗) → (◡𝐺‘𝑚) = (◡𝐺‘(𝐺‘𝑗)))
14	13	fveqeq2d 5594	. . . . . . . . . 10 ⊢ (𝑚 = (𝐺‘𝑗) → ((𝑦‘(◡𝐺‘𝑚)) = ∅ ↔ (𝑦‘(◡𝐺‘(𝐺‘𝑗))) = ∅))
15		simprl 529	. . . . . . . . . . 11 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝑗 ∈ ω)
16	11, 1, 15	frec2uzuzd 10560	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺‘𝑗) ∈ (ℤ≥‘0))
17	11, 1	frec2uzf1od 10564	. . . . . . . . . . . . 13 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝐺:ω–1-1-onto→(ℤ≥‘0))
18		f1ocnvfv1 5856	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑗 ∈ ω) → (◡𝐺‘(𝐺‘𝑗)) = 𝑗)
19	17, 15, 18	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (◡𝐺‘(𝐺‘𝑗)) = 𝑗)
20	19	fveq2d 5590	. . . . . . . . . . 11 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑦‘(◡𝐺‘(𝐺‘𝑗))) = (𝑦‘𝑗))
21		simprr 531	. . . . . . . . . . 11 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑦‘𝑗) = ∅)
22	20, 21	eqtrd 2239	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑦‘(◡𝐺‘(𝐺‘𝑗))) = ∅)
23	14, 16, 22	elrabd 2933	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺‘𝑗) ∈ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅})
24		nninff 7236	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
25		2ssom 6620	. . . . . . . . . . . . . 14 ⊢ 2o ⊆ ω
26	25	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ∞ → 2o ⊆ ω)
27	24, 26	fssd 5445	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶ω)
28	27	ad3antlr 493	. . . . . . . . . . 11 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑚 ∈ (0...(𝐺‘𝑗))) → 𝑦:ω⟶ω)
29		elfzuz 10156	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝐺‘𝑗)) → 𝑚 ∈ (ℤ≥‘0))
30		f1ocnvdm 5860	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑚) ∈ ω)
31	17, 29, 30	syl2an 289	. . . . . . . . . . 11 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑚 ∈ (0...(𝐺‘𝑗))) → (◡𝐺‘𝑚) ∈ ω)
32	28, 31	ffvelcdmd 5726	. . . . . . . . . 10 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑚 ∈ (0...(𝐺‘𝑗))) → (𝑦‘(◡𝐺‘𝑚)) ∈ ω)
33		peano1 4647	. . . . . . . . . 10 ⊢ ∅ ∈ ω
34		nndceq 6595	. . . . . . . . . 10 ⊢ (((𝑦‘(◡𝐺‘𝑚)) ∈ ω ∧ ∅ ∈ ω) → DECID (𝑦‘(◡𝐺‘𝑚)) = ∅)
35	32, 33, 34	sylancl 413	. . . . . . . . 9 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑚 ∈ (0...(𝐺‘𝑗))) → DECID (𝑦‘(◡𝐺‘𝑚)) = ∅)
36	11, 12, 23, 35	infssuzcldc 10391	. . . . . . . 8 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅})
37	10, 36	sselid 3193	. . . . . . 7 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ ℕ0*)
38	24	adantl 277	. . . . . . . . . 10 ⊢ ((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) → 𝑦:ω⟶2o)
39	38	adantr 276	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝑦:ω⟶2o)
40	39	ffnd 5433	. . . . . . . 8 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝑦 Fn ω)
41	4	a1i 9	. . . . . . . . . . 11 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝐼:ℕ0*⟶ℕ∞)
42	41, 37	ffvelcdmd 5726	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ℕ∞)
43		nninff 7236	. . . . . . . . . 10 ⊢ ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ℕ∞ → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )):ω⟶2o)
44	42, 43	syl 14	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )):ω⟶2o)
45	44	ffnd 5433	. . . . . . . 8 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) Fn ω)
46		2fveq3 5591	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘𝑚)))
47	46	eqeq1d 2215	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘𝑚)) = ∅))
48	47	cbvrabv 2772	. . . . . . . . . . . . 13 ⊢ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} = {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}
49	48	infeq1i 7127	. . . . . . . . . . . 12 ⊢ inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )
50	49	fveq2i 5589	. . . . . . . . . . 11 ⊢ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
51	50	eleq2i 2273	. . . . . . . . . 10 ⊢ (𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) ↔ 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
52		simpr 110	. . . . . . . . . . . . 13 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )))
53	52, 51	sylib 122	. . . . . . . . . . . 12 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
54	53	iftrued 3580	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅) = 1o)
55	3	fveq1i 5587	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
56	6, 36	sselid 3193	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0))
57	56, 7	eleqtrrdi 2300	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ ℕ0)
58	57	nn0nepnfd 9381	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ≠ +∞)
59	58	necomd 2463	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → +∞ ≠ inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
60		fvunsng 5788	. . . . . . . . . . . . . . . . . 18 ⊢ ((inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ ℕ0 ∧ +∞ ≠ inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) → (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = ((𝐹 ∘ ◡𝐺)‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
61	57, 59, 60	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = ((𝐹 ∘ ◡𝐺)‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
62	55, 61	eqtrid 2251	. . . . . . . . . . . . . . . 16 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = ((𝐹 ∘ ◡𝐺)‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
63		dff1o4 5539	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘0) ↔ (𝐺 Fn ω ∧ ◡𝐺 Fn (ℤ≥‘0)))
64	17, 63	sylib 122	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺 Fn ω ∧ ◡𝐺 Fn (ℤ≥‘0)))
65	64	simprd 114	. . . . . . . . . . . . . . . . 17 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → ◡𝐺 Fn (ℤ≥‘0))
66		fvco2 5658	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐺 Fn (ℤ≥‘0) ∧ inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0)) → ((𝐹 ∘ ◡𝐺)‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = (𝐹‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))))
67	65, 56, 66	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → ((𝐹 ∘ ◡𝐺)‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = (𝐹‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))))
68		eleq2 2270	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))))
69	68	ifbid 3594	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
70	69	mpteq2dv 4140	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅)))
71		f1ocnvdm 5860	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0)) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ω)
72	17, 56, 71	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ω)
73		omex 4646	. . . . . . . . . . . . . . . . . . 19 ⊢ ω ∈ V
74	73	mptex 5820	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅)) ∈ V
75	74	a1i 9	. . . . . . . . . . . . . . . . 17 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅)) ∈ V)
76	2, 70, 72, 75	fvmptd3 5683	. . . . . . . . . . . . . . . 16 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐹‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) = (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅)))
77	62, 67, 76	3eqtrd 2243	. . . . . . . . . . . . . . 15 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) = (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅)))
78	77	fveq1d 5588	. . . . . . . . . . . . . 14 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))‘𝑘))
79	78	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))‘𝑘))
80		eqid 2206	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
81		eleq1w 2267	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ↔ 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))))
82	81	ifbid 3594	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅) = if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
83		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → 𝑘 ∈ ω)
84		1lt2o 6538	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ 2o
85	84	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
86		0lt2o 6537	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ 2o
87	86	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ∅ ∈ 2o)
88	72	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ω)
89		nndcel 6596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ω) → DECID 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
90	83, 88, 89	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → DECID 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
91	85, 87, 90	ifcldcd 3610	. . . . . . . . . . . . . 14 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅) ∈ 2o)
92	80, 82, 83, 91	fvmptd3 5683	. . . . . . . . . . . . 13 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))‘𝑘) = if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
93	79, 92	eqtrd 2239	. . . . . . . . . . . 12 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘) = if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
94	93	adantr 276	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘) = if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
95		0zd 9397	. . . . . . . . . . . . . . . . . 18 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → 0 ∈ ℤ)
96		simplr 528	. . . . . . . . . . . . . . . . . 18 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → 𝑘 ∈ ω)
97	50, 88	eqeltrid 2293	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) ∈ ω)
98	97	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) ∈ ω)
99	95, 1, 96, 98	frec2uzlt2d 10562	. . . . . . . . . . . . . . . . 17 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) ↔ (𝐺‘𝑘) < (𝐺‘(◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )))))
100	52, 99	mpbid 147	. . . . . . . . . . . . . . . 16 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝐺‘𝑘) < (𝐺‘(◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))))
101	17	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → 𝐺:ω–1-1-onto→(ℤ≥‘0))
102	49, 56	eqeltrid 2293	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0))
103	102	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0))
104		f1ocnvfv2 5857	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0)) → (𝐺‘(◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) = inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))
105	101, 103, 104	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝐺‘(◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) = inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))
106	100, 105	breqtrd 4074	. . . . . . . . . . . . . . 15 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝐺‘𝑘) < inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))
107	49, 57	eqeltrid 2293	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ∈ ℕ0)
108	107	ad3antrrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ∈ ℕ0)
109	108	nn0red 9362	. . . . . . . . . . . . . . . . 17 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ∈ ℝ)
110		elrabi 2928	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} → (𝐺‘𝑘) ∈ (ℤ≥‘0))
111	110, 7	eleqtrrdi 2300	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} → (𝐺‘𝑘) ∈ ℕ0)
112	111	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → (𝐺‘𝑘) ∈ ℕ0)
113	112	nn0red 9362	. . . . . . . . . . . . . . . . 17 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → (𝐺‘𝑘) ∈ ℝ)
114		0zd 9397	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → 0 ∈ ℤ)
115		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅})
116	39	ad4antr 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → 𝑦:ω⟶2o)
117	17	ad4antr 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → 𝐺:ω–1-1-onto→(ℤ≥‘0))
118		elfzuz 10156	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (0...(𝐺‘𝑘)) → 𝑚 ∈ (ℤ≥‘0))
119	118	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → 𝑚 ∈ (ℤ≥‘0))
120	117, 119, 30	syl2anc 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → (◡𝐺‘𝑚) ∈ ω)
121	116, 120	ffvelcdmd 5726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → (𝑦‘(◡𝐺‘𝑚)) ∈ 2o)
122	25, 121	sselid 3193	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → (𝑦‘(◡𝐺‘𝑚)) ∈ ω)
123	122, 33, 34	sylancl 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) ∧ 𝑚 ∈ (0...(𝐺‘𝑘))) → DECID (𝑦‘(◡𝐺‘𝑚)) = ∅)
124	114, 48, 115, 123	infssuzledc 10390	. . . . . . . . . . . . . . . . 17 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) ≤ (𝐺‘𝑘))
125	109, 113, 124	lensymd 8207	. . . . . . . . . . . . . . . 16 ⊢ ((((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) ∧ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}) → ¬ (𝐺‘𝑘) < inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))
126	125	ex 115	. . . . . . . . . . . . . . 15 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} → ¬ (𝐺‘𝑘) < inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )))
127	106, 126	mt2d 626	. . . . . . . . . . . . . 14 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ¬ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅})
128		2fveq3 5591	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝐺‘𝑘) → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘(𝐺‘𝑘))))
129	128	eqeq1d 2215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝐺‘𝑘) → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
130	129	elrab 2931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ ((𝐺‘𝑘) ∈ (ℤ≥‘0) ∧ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
131		f1of 5531	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘0) → 𝐺:ω⟶(ℤ≥‘0))
132	17, 131	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝐺:ω⟶(ℤ≥‘0))
133	132	ffvelcdmda 5725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝐺‘𝑘) ∈ (ℤ≥‘0))
134	133	biantrurd 305	. . . . . . . . . . . . . . . . 17 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅ ↔ ((𝐺‘𝑘) ∈ (ℤ≥‘0) ∧ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅)))
135	130, 134	bitr4id 199	. . . . . . . . . . . . . . . 16 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
136	135	notbid 669	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (¬ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ ¬ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
137	136	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (¬ (𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ ¬ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
138	127, 137	mpbid 147	. . . . . . . . . . . . 13 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ¬ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅)
139		f1ocnvfv1 5856	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑘 ∈ ω) → (◡𝐺‘(𝐺‘𝑘)) = 𝑘)
140	17, 139	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (◡𝐺‘(𝐺‘𝑘)) = 𝑘)
141	140	fveq2d 5590	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = (𝑦‘𝑘))
142	141	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = (𝑦‘𝑘))
143	142	eqeq1d 2215	. . . . . . . . . . . . 13 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ((𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅ ↔ (𝑦‘𝑘) = ∅))
144	138, 143	mtbid 674	. . . . . . . . . . . 12 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ¬ (𝑦‘𝑘) = ∅)
145	39	ffvelcdmda 5725	. . . . . . . . . . . . . . . 16 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) ∈ 2o)
146		df2o3 6526	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
147	145, 146	eleqtrdi 2299	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) ∈ {∅, 1o})
148		elpri 3658	. . . . . . . . . . . . . . 15 ⊢ ((𝑦‘𝑘) ∈ {∅, 1o} → ((𝑦‘𝑘) = ∅ ∨ (𝑦‘𝑘) = 1o))
149	147, 148	syl 14	. . . . . . . . . . . . . 14 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝑦‘𝑘) = ∅ ∨ (𝑦‘𝑘) = 1o))
150	149	orcomd 731	. . . . . . . . . . . . 13 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝑦‘𝑘) = 1o ∨ (𝑦‘𝑘) = ∅))
151	150	adantr 276	. . . . . . . . . . . 12 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → ((𝑦‘𝑘) = 1o ∨ (𝑦‘𝑘) = ∅))
152	144, 151	ecased 1362	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝑦‘𝑘) = 1o)
153	54, 94, 152	3eqtr4rd 2250	. . . . . . . . . 10 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ))) → (𝑦‘𝑘) = ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘))
154	51, 153	sylan2br 288	. . . . . . . . 9 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) → (𝑦‘𝑘) = ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘))
155		ssnel 4622	. . . . . . . . . . . 12 ⊢ ((◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘 → ¬ 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
156	155	adantl 277	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → ¬ 𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
157	156	iffalsed 3583	. . . . . . . . . 10 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅) = ∅)
158	93	adantr 276	. . . . . . . . . 10 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘) = if(𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )), 1o, ∅))
159		simp-4r 542	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → 𝑦 ∈ ℕ∞)
160	72	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ω)
161		simplr 528	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → 𝑘 ∈ ω)
162		simpr 110	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘)
163	36, 48	eleqtrrdi 2300	. . . . . . . . . . . . . 14 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅})
164		2fveq3 5591	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))))
165	164	eqeq1d 2215	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) = ∅))
166	165	elrab 2931	. . . . . . . . . . . . . 14 ⊢ (inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ (inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0) ∧ (𝑦‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) = ∅))
167	163, 166	sylib 122	. . . . . . . . . . . . 13 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0) ∧ (𝑦‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) = ∅))
168	167	simprd 114	. . . . . . . . . . . 12 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑦‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) = ∅)
169	168	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → (𝑦‘(◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) = ∅)
170	159, 160, 161, 162, 169	nninfninc 7237	. . . . . . . . . 10 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → (𝑦‘𝑘) = ∅)
171	157, 158, 170	3eqtr4rd 2250	. . . . . . . . 9 ⊢ (((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘) → (𝑦‘𝑘) = ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘))
172		nntri3or 6589	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ω ∧ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ω) → (𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ 𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘))
173	83, 88, 172	syl2anc 411	. . . . . . . . . . 11 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ 𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘))
174		3orass 984	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ 𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘) ↔ (𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘)))
175	173, 174	sylib 122	. . . . . . . . . 10 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘)))
176		eqimss2 3250	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘)
177	176	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → (𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘))
178		nnon 4663	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → 𝑘 ∈ On)
179		onelss 4439	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ On → ((◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘 → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘))
180	178, 179	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → ((◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘 → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘))
181	177, 180	jaod 719	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → ((𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘))
182	181	adantl 277	. . . . . . . . . . 11 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘) → (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘))
183	182	orim2d 790	. . . . . . . . . 10 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → ((𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (𝑘 = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ 𝑘)) → (𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘)))
184	175, 183	mpd 13	. . . . . . . . 9 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑘 ∈ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∨ (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ⊆ 𝑘))
185	154, 171, 184	mpjaodan 800	. . . . . . . 8 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = ((𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))‘𝑘))
186	40, 45, 185	eqfnfvd 5690	. . . . . . 7 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝑦 = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
187		fveq2 5586	. . . . . . . 8 ⊢ (𝑧 = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) → (𝐼‘𝑧) = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
188	187	rspceeqv 2897	. . . . . . 7 ⊢ ((inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ ℕ0* ∧ 𝑦 = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
189	37, 186, 188	syl2anc 411	. . . . . 6 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
190	189	rexlimdvaa 2625	. . . . 5 ⊢ ((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) → (∃𝑗 ∈ ω (𝑦‘𝑗) = ∅ → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧)))
191	190	imp 124	. . . 4 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∃𝑗 ∈ ω (𝑦‘𝑗) = ∅) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
192		pnf0xnn0 9378	. . . . 5 ⊢ +∞ ∈ ℕ0*
193	24	ffnd 5433	. . . . . . 7 ⊢ (𝑦 ∈ ℕ∞ → 𝑦 Fn ω)
194	193	ad2antlr 489	. . . . . 6 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → 𝑦 Fn ω)
195		1oex 6520	. . . . . . . 8 ⊢ 1o ∈ V
196	1, 2, 3	inftonninf 10600	. . . . . . . 8 ⊢ (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o)
197	195, 196	fnmpti 5411	. . . . . . 7 ⊢ (𝐼‘+∞) Fn ω
198	197	a1i 9	. . . . . 6 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → (𝐼‘+∞) Fn ω)
199		fveqeq2 5595	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑦‘𝑗) = 1o ↔ (𝑦‘𝑘) = 1o))
200		simplr 528	. . . . . . . 8 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o)
201		simpr 110	. . . . . . . 8 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → 𝑘 ∈ ω)
202	199, 200, 201	rspcdva 2884	. . . . . . 7 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = 1o)
203		eqidd 2207	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → 1o = 1o)
204	203, 196, 195	fvmpt 5666	. . . . . . . 8 ⊢ (𝑘 ∈ ω → ((𝐼‘+∞)‘𝑘) = 1o)
205	204	adantl 277	. . . . . . 7 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → ((𝐼‘+∞)‘𝑘) = 1o)
206	202, 205	eqtr4d 2242	. . . . . 6 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = ((𝐼‘+∞)‘𝑘))
207	194, 198, 206	eqfnfvd 5690	. . . . 5 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → 𝑦 = (𝐼‘+∞))
208		fveq2 5586	. . . . . 6 ⊢ (𝑧 = +∞ → (𝐼‘𝑧) = (𝐼‘+∞))
209	208	rspceeqv 2897	. . . . 5 ⊢ ((+∞ ∈ ℕ0* ∧ 𝑦 = (𝐼‘+∞)) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
210	192, 207, 209	sylancr 414	. . . 4 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
211		isomni 7250	. . . . . . . 8 ⊢ (ω ∈ V → (ω ∈ Omni ↔ ∀𝑦(𝑦:ω⟶2o → (∃𝑗 ∈ ω (𝑦‘𝑗) = ∅ ∨ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o))))
212	73, 211	ax-mp 5	. . . . . . 7 ⊢ (ω ∈ Omni ↔ ∀𝑦(𝑦:ω⟶2o → (∃𝑗 ∈ ω (𝑦‘𝑗) = ∅ ∨ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o)))
213	212	biimpi 120	. . . . . 6 ⊢ (ω ∈ Omni → ∀𝑦(𝑦:ω⟶2o → (∃𝑗 ∈ ω (𝑦‘𝑗) = ∅ ∨ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o)))
214	213	19.21bi 1582	. . . . 5 ⊢ (ω ∈ Omni → (𝑦:ω⟶2o → (∃𝑗 ∈ ω (𝑦‘𝑗) = ∅ ∨ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o)))
215	214, 24	impel 280	. . . 4 ⊢ ((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) → (∃𝑗 ∈ ω (𝑦‘𝑗) = ∅ ∨ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o))
216	191, 210, 215	mpjaodan 800	. . 3 ⊢ ((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
217	216	ralrimiva 2580	. 2 ⊢ (ω ∈ Omni → ∀𝑦 ∈ ℕ∞ ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
218		dffo3 5737	. 2 ⊢ (𝐼:ℕ0*–onto→ℕ∞ ↔ (𝐼:ℕ0*⟶ℕ∞ ∧ ∀𝑦 ∈ ℕ∞ ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧)))
219	5, 217, 218	sylanbrc 417	1 ⊢ (ω ∈ Omni → 𝐼:ℕ0*–onto→ℕ∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836   ∨ w3o 980  ∀wal 1371   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∀wral 2485  ∃wrex 2486  {crab 2489  Vcvv 2773   ∪ cun 3166   ⊆ wss 3168  ∅c0 3462  ifcif 3573  {csn 3635  {cpr 3636  ⟨cop 3638   class class class wbr 4048   ↦ cmpt 4110  Oncon0 4415  ωcom 4643   × cxp 4678  ^◡ccnv 4679   ∘ ccom 4684   Fn wfn 5272  ⟶wf 5273  –onto→wfo 5275  –1-1-onto→wf1o 5276  ‘cfv 5277  (class class class)co 5954  freccfrec 6486  1_oc1o 6505  2_oc2o 6506  infcinf 7097  ℕ_∞xnninf 7233  Omnicomni 7248  ℝcr 7937  0cc0 7938  1c1 7939   + caddc 7941  +∞cpnf 8117   < clt 8120  ℕ₀cn0 9308  ℕ₀^*cxnn0 9371  ℤcz 9385  ℤ_≥cuz 9661  ...cfz 10143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-addass 8040  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-ilim 4421  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-isom 5286  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-frec 6487  df-1o 6512  df-2o 6513  df-map 6747  df-sup 7098  df-inf 7099  df-nninf 7234  df-omni 7249  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-inn 9050  df-n0 9309  df-xnn0 9372  df-z 9386  df-uz 9662  df-fz 10144  df-fzo 10278

This theorem is referenced by:  nninfct  12412