Theorem nninfctlemfo 12207

Description: Lemma for nninfct 12208. (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 10531	. . 3 ⊢ 𝐼:ℕ0*⟶ℕ∞
5	4	a1i 9	. 2 ⊢ (ω ∈ Omni → 𝐼:ℕ0*⟶ℕ∞)
6		ssrab2 3268	. . . . . . . . 9 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} ⊆ (ℤ≥‘0)
7		nn0uz 9636	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
8		nn0ssxnn0 9315	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℕ0*
9	7, 8	eqsstrri 3216	. . . . . . . . 9 ⊢ (ℤ≥‘0) ⊆ ℕ0*
10	6, 9	sstri 3192	. . . . . . . 8 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} ⊆ ℕ0*
11		0zd 9338	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 0 ∈ ℤ)
12		eqid 2196	. . . . . . . . 9 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} = {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}
13		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑗) → (◡𝐺‘𝑚) = (◡𝐺‘(𝐺‘𝑗)))
14	13	fveqeq2d 5566	. . . . . . . . . 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 10494	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺‘𝑗) ∈ (ℤ≥‘0))
17	11, 1	frec2uzf1od 10498	. . . . . . . . . . . . 13 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝐺:ω–1-1-onto→(ℤ≥‘0))
18		f1ocnvfv1 5824	. . . . . . . . . . . . 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 5562	. . . . . . . . . . 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 2229	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑦‘(◡𝐺‘(𝐺‘𝑗))) = ∅)
23	14, 16, 22	elrabd 2922	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺‘𝑗) ∈ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅})
24		nninff 7188	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
25		2ssom 6582	. . . . . . . . . . . . . 14 ⊢ 2o ⊆ ω
26	25	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ∞ → 2o ⊆ ω)
27	24, 26	fssd 5420	. . . . . . . . . . . 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 10096	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝐺‘𝑗)) → 𝑚 ∈ (ℤ≥‘0))
30		f1ocnvdm 5828	. . . . . . . . . . . 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 5698	. . . . . . . . . 10 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑚 ∈ (0...(𝐺‘𝑗))) → (𝑦‘(◡𝐺‘𝑚)) ∈ ω)
33		peano1 4630	. . . . . . . . . 10 ⊢ ∅ ∈ ω
34		nndceq 6557	. . . . . . . . . 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 10325	. . . . . . . 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 3181	. . . . . . 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 5408	. . . . . . . 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 5698	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ℕ∞)
43		nninff 7188	. . . . . . . . . 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 5408	. . . . . . . 8 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) Fn ω)
46		2fveq3 5563	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘𝑚)))
47	46	eqeq1d 2205	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘𝑚)) = ∅))
48	47	cbvrabv 2762	. . . . . . . . . . . . 13 ⊢ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} = {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}
49	48	infeq1i 7079	. . . . . . . . . . . 12 ⊢ inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )
50	49	fveq2i 5561	. . . . . . . . . . 11 ⊢ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
51	50	eleq2i 2263	. . . . . . . . . 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 3568	. . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . . 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 3181	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0))
57	56, 7	eleqtrrdi 2290	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ ℕ0)
58	57	nn0nepnfd 9322	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ≠ +∞)
59	58	necomd 2453	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → +∞ ≠ inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
60		fvunsng 5756	. . . . . . . . . . . . . . . . . 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 2241	. . . . . . . . . . . . . . . 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 5512	. . . . . . . . . . . . . . . . . . 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 5630	. . . . . . . . . . . . . . . . 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 2260	. . . . . . . . . . . . . . . . . . 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 3582	. . . . . . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . . . 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 5828	. . . . . . . . . . . . . . . . . 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 4629	. . . . . . . . . . . . . . . . . . 19 ⊢ ω ∈ V
74	73	mptex 5788	. . . . . . . . . . . . . . . . . 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 5655	. . . . . . . . . . . . . . . 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 2233	. . . . . . . . . . . . . . 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 5560	. . . . . . . . . . . . . 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 2196	. . . . . . . . . . . . . 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 2257	. . . . . . . . . . . . . . 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 3582	. . . . . . . . . . . . . 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 6500	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ 2o
85	84	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
86		0lt2o 6499	. . . . . . . . . . . . . . . 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 6558	. . . . . . . . . . . . . . . 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 3597	. . . . . . . . . . . . . 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 5655	. . . . . . . . . . . . 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 2229	. . . . . . . . . . . 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 9338	. . . . . . . . . . . . . . . . . 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 2283	. . . . . . . . . . . . . . . . . . 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 10496	. . . . . . . . . . . . . . . . 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 2283	. . . . . . . . . . . . . . . . . 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 5825	. . . . . . . . . . . . . . . . 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 4059	. . . . . . . . . . . . . . 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 2283	. . . . . . . . . . . . . . . . . . 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 9303	. . . . . . . . . . . . . . . . 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 2917	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} → (𝐺‘𝑘) ∈ (ℤ≥‘0))
111	110, 7	eleqtrrdi 2290	. . . . . . . . . . . . . . . . . . 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 9303	. . . . . . . . . . . . . . . . 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 9338	. . . . . . . . . . . . . . . . . 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 10096	. . . . . . . . . . . . . . . . . . . . . . 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 5698	. . . . . . . . . . . . . . . . . . . 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 3181	. . . . . . . . . . . . . . . . . . 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 10324	. . . . . . . . . . . . . . . . 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 8148	. . . . . . . . . . . . . . . 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 5563	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝐺‘𝑘) → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘(𝐺‘𝑘))))
129	128	eqeq1d 2205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝐺‘𝑘) → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
130	129	elrab 2920	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ ((𝐺‘𝑘) ∈ (ℤ≥‘0) ∧ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
131		f1of 5504	. . . . . . . . . . . . . . . . . . . 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 5697	. . . . . . . . . . . . . . . . . 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 668	. . . . . . . . . . . . . . 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 5824	. . . . . . . . . . . . . . . . 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 5562	. . . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . 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 673	. . . . . . . . . . . 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 5697	. . . . . . . . . . . . . . . 16 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) ∈ 2o)
146		df2o3 6488	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
147	145, 146	eleqtrdi 2289	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) ∈ {∅, 1o})
148		elpri 3645	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 1360	. . . . . . . . . . 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 2240	. . . . . . . . . 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 4605	. . . . . . . . . . . 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 3571	. . . . . . . . . 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 2290	. . . . . . . . . . . . . 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 5563	. . . . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . . . 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 2920	. . . . . . . . . . . . . 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 7189	. . . . . . . . . 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 2240	. . . . . . . . 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 6551	. . . . . . . . . . . 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 983	. . . . . . . . . . 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 3238	. . . . . . . . . . . . . 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 4646	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → 𝑘 ∈ On)
179		onelss 4422	. . . . . . . . . . . . . 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 718	. . . . . . . . . . . 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 789	. . . . . . . . . 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 799	. . . . . . . 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 5662	. . . . . . 7 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝑦 = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
187		fveq2 5558	. . . . . . . 8 ⊢ (𝑧 = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) → (𝐼‘𝑧) = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
188	187	rspceeqv 2886	. . . . . . 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 2615	. . . . 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 9319	. . . . 5 ⊢ +∞ ∈ ℕ0*
193	24	ffnd 5408	. . . . . . 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 6482	. . . . . . . 8 ⊢ 1o ∈ V
196	1, 2, 3	inftonninf 10534	. . . . . . . 8 ⊢ (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o)
197	195, 196	fnmpti 5386	. . . . . . 7 ⊢ (𝐼‘+∞) Fn ω
198	197	a1i 9	. . . . . 6 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → (𝐼‘+∞) Fn ω)
199		fveqeq2 5567	. . . . . . . 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 2873	. . . . . . 7 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = 1o)
203		eqidd 2197	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → 1o = 1o)
204	203, 196, 195	fvmpt 5638	. . . . . . . 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 2232	. . . . . 6 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = ((𝐼‘+∞)‘𝑘))
207	194, 198, 206	eqfnfvd 5662	. . . . 5 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → 𝑦 = (𝐼‘+∞))
208		fveq2 5558	. . . . . 6 ⊢ (𝑧 = +∞ → (𝐼‘𝑧) = (𝐼‘+∞))
209	208	rspceeqv 2886	. . . . 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 7202	. . . . . . . 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 1572	. . . . 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 799	. . 3 ⊢ ((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) → ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
217	216	ralrimiva 2570	. 2 ⊢ (ω ∈ Omni → ∀𝑦 ∈ ℕ∞ ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
218		dffo3 5709	. 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 709  DECID wdc 835   ∨ w3o 979  ∀wal 1362   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ∃wrex 2476  {crab 2479  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157  ∅c0 3450  ifcif 3561  {csn 3622  {cpr 3623  ⟨cop 3625   class class class wbr 4033   ↦ cmpt 4094  Oncon0 4398  ωcom 4626   × cxp 4661  ^◡ccnv 4662   ∘ ccom 4667   Fn wfn 5253  ⟶wf 5254  –onto→wfo 5256  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  freccfrec 6448  1_oc1o 6467  2_oc2o 6468  infcinf 7049  ℕ_∞xnninf 7185  Omnicomni 7200  ℝcr 7878  0cc0 7879  1c1 7880   + caddc 7882  +∞cpnf 8058   < clt 8061  ℕ₀cn0 9249  ℕ₀^*cxnn0 9312  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-map 6709  df-sup 7050  df-inf 7051  df-nninf 7186  df-omni 7201  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-xnn0 9313  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218

This theorem is referenced by:  nninfct  12208