Theorem nninfctlemfo 12177

Description: Lemma for nninfct 12178. (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 10510	. . 3 ⊢ 𝐼:ℕ0*⟶ℕ∞
5	4	a1i 9	. 2 ⊢ (ω ∈ Omni → 𝐼:ℕ0*⟶ℕ∞)
6		ssrab2 3264	. . . . . . . . 9 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} ⊆ (ℤ≥‘0)
7		nn0uz 9627	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
8		nn0ssxnn0 9306	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℕ0*
9	7, 8	eqsstrri 3212	. . . . . . . . 9 ⊢ (ℤ≥‘0) ⊆ ℕ0*
10	6, 9	sstri 3188	. . . . . . . 8 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} ⊆ ℕ0*
11		0zd 9329	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 0 ∈ ℤ)
12		eqid 2193	. . . . . . . . 9 ⊢ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅} = {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}
13		fveq2 5554	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑗) → (◡𝐺‘𝑚) = (◡𝐺‘(𝐺‘𝑗)))
14	13	fveqeq2d 5562	. . . . . . . . . 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 10473	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺‘𝑗) ∈ (ℤ≥‘0))
17	11, 1	frec2uzf1od 10477	. . . . . . . . . . . . 13 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝐺:ω–1-1-onto→(ℤ≥‘0))
18		f1ocnvfv1 5820	. . . . . . . . . . . . 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 5558	. . . . . . . . . . 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 2226	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝑦‘(◡𝐺‘(𝐺‘𝑗))) = ∅)
23	14, 16, 22	elrabd 2918	. . . . . . . . 9 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐺‘𝑗) ∈ {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅})
24		nninff 7181	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
25		2ssom 6577	. . . . . . . . . . . . . 14 ⊢ 2o ⊆ ω
26	25	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ∞ → 2o ⊆ ω)
27	24, 26	fssd 5416	. . . . . . . . . . . 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 10087	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝐺‘𝑗)) → 𝑚 ∈ (ℤ≥‘0))
30		f1ocnvdm 5824	. . . . . . . . . . . 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 5694	. . . . . . . . . 10 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑚 ∈ (0...(𝐺‘𝑗))) → (𝑦‘(◡𝐺‘𝑚)) ∈ ω)
33		peano1 4626	. . . . . . . . . 10 ⊢ ∅ ∈ ω
34		nndceq 6552	. . . . . . . . . 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 12088	. . . . . . . 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 3177	. . . . . . 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 5404	. . . . . . . 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 5694	. . . . . . . . . 10 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) ∈ ℕ∞)
43		nninff 7181	. . . . . . . . . 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 5404	. . . . . . . 8 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )) Fn ω)
46		2fveq3 5559	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘𝑚)))
47	46	eqeq1d 2202	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘𝑚)) = ∅))
48	47	cbvrabv 2759	. . . . . . . . . . . . 13 ⊢ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} = {𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}
49	48	infeq1i 7072	. . . . . . . . . . . 12 ⊢ inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < ) = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )
50	49	fveq2i 5557	. . . . . . . . . . 11 ⊢ (◡𝐺‘inf({𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅}, ℝ, < )) = (◡𝐺‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
51	50	eleq2i 2260	. . . . . . . . . 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 3564	. . . . . . . . . . 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 5555	. . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ (ℤ≥‘0))
57	56, 7	eleqtrrdi 2287	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ∈ ℕ0)
58	57	nn0nepnfd 9313	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) ≠ +∞)
59	58	necomd 2450	. . . . . . . . . . . . . . . . . 18 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → +∞ ≠ inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ))
60		fvunsng 5752	. . . . . . . . . . . . . . . . . 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 2238	. . . . . . . . . . . . . . . 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 5508	. . . . . . . . . . . . . . . . . . 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 5626	. . . . . . . . . . . . . . . . 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 2257	. . . . . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . . . . 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 5824	. . . . . . . . . . . . . . . . . 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 4625	. . . . . . . . . . . . . . . . . . 19 ⊢ ω ∈ V
74	73	mptex 5784	. . . . . . . . . . . . . . . . . 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 5651	. . . . . . . . . . . . . . . 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 2230	. . . . . . . . . . . . . . 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 5556	. . . . . . . . . . . . . 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 2193	. . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . 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 6495	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ 2o
85	84	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
86		0lt2o 6494	. . . . . . . . . . . . . . . 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 6553	. . . . . . . . . . . . . . . 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 3593	. . . . . . . . . . . . . 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 5651	. . . . . . . . . . . . 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 2226	. . . . . . . . . . . 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 9329	. . . . . . . . . . . . . . . . . 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 2280	. . . . . . . . . . . . . . . . . . 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 10475	. . . . . . . . . . . . . . . . 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 2280	. . . . . . . . . . . . . . . . . 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 5821	. . . . . . . . . . . . . . . . 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 4055	. . . . . . . . . . . . . . 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 2280	. . . . . . . . . . . . . . . . . . 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 9294	. . . . . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} → (𝐺‘𝑘) ∈ (ℤ≥‘0))
111	110, 7	eleqtrrdi 2287	. . . . . . . . . . . . . . . . . . 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 9294	. . . . . . . . . . . . . . . . 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 9329	. . . . . . . . . . . . . . . . . 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 10087	. . . . . . . . . . . . . . . . . . . . . . 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 5694	. . . . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . . 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 12087	. . . . . . . . . . . . . . . . 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 8141	. . . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝐺‘𝑘) → (𝑦‘(◡𝐺‘𝑛)) = (𝑦‘(◡𝐺‘(𝐺‘𝑘))))
129	128	eqeq1d 2202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝐺‘𝑘) → ((𝑦‘(◡𝐺‘𝑛)) = ∅ ↔ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
130	129	elrab 2916	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑘) ∈ {𝑛 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑛)) = ∅} ↔ ((𝐺‘𝑘) ∈ (ℤ≥‘0) ∧ (𝑦‘(◡𝐺‘(𝐺‘𝑘))) = ∅))
131		f1of 5500	. . . . . . . . . . . . . . . . . . . 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 5693	. . . . . . . . . . . . . . . . . 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 5820	. . . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . 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 2202	. . . . . . . . . . . . 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 5693	. . . . . . . . . . . . . . . 16 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) ∈ 2o)
146		df2o3 6483	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
147	145, 146	eleqtrdi 2286	. . . . . . . . . . . . . . 15 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) ∈ {∅, 1o})
148		elpri 3641	. . . . . . . . . . . . . . 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 2237	. . . . . . . . . 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 4601	. . . . . . . . . . . 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 3567	. . . . . . . . . 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 2287	. . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . 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 2202	. . . . . . . . . . . . . . 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 2916	. . . . . . . . . . . . . 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 7182	. . . . . . . . . 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 2237	. . . . . . . . 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 6546	. . . . . . . . . . . 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 3234	. . . . . . . . . . . . . 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 4642	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → 𝑘 ∈ On)
179		onelss 4418	. . . . . . . . . . . . . 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 5658	. . . . . . 7 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ (𝑗 ∈ ω ∧ (𝑦‘𝑗) = ∅)) → 𝑦 = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
187		fveq2 5554	. . . . . . . 8 ⊢ (𝑧 = inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < ) → (𝐼‘𝑧) = (𝐼‘inf({𝑚 ∈ (ℤ≥‘0) ∣ (𝑦‘(◡𝐺‘𝑚)) = ∅}, ℝ, < )))
188	187	rspceeqv 2882	. . . . . . 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 2612	. . . . 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 9310	. . . . 5 ⊢ +∞ ∈ ℕ0*
193	24	ffnd 5404	. . . . . . 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 6477	. . . . . . . 8 ⊢ 1o ∈ V
196	1, 2, 3	inftonninf 10513	. . . . . . . 8 ⊢ (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o)
197	195, 196	fnmpti 5382	. . . . . . 7 ⊢ (𝐼‘+∞) Fn ω
198	197	a1i 9	. . . . . 6 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → (𝐼‘+∞) Fn ω)
199		fveqeq2 5563	. . . . . . . 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 2869	. . . . . . 7 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = 1o)
203		eqidd 2194	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → 1o = 1o)
204	203, 196, 195	fvmpt 5634	. . . . . . . 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 2229	. . . . . 6 ⊢ ((((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) ∧ 𝑘 ∈ ω) → (𝑦‘𝑘) = ((𝐼‘+∞)‘𝑘))
207	194, 198, 206	eqfnfvd 5658	. . . . 5 ⊢ (((ω ∈ Omni ∧ 𝑦 ∈ ℕ∞) ∧ ∀𝑗 ∈ ω (𝑦‘𝑗) = 1o) → 𝑦 = (𝐼‘+∞))
208		fveq2 5554	. . . . . 6 ⊢ (𝑧 = +∞ → (𝐼‘𝑧) = (𝐼‘+∞))
209	208	rspceeqv 2882	. . . . 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 7195	. . . . . . . 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 1569	. . . . 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 2567	. 2 ⊢ (ω ∈ Omni → ∀𝑦 ∈ ℕ∞ ∃𝑧 ∈ ℕ0* 𝑦 = (𝐼‘𝑧))
218		dffo3 5705	. 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 2164   ≠ wne 2364  ∀wral 2472  ∃wrex 2473  {crab 2476  Vcvv 2760   ∪ cun 3151   ⊆ wss 3153  ∅c0 3446  ifcif 3557  {csn 3618  {cpr 3619  ⟨cop 3621   class class class wbr 4029   ↦ cmpt 4090  Oncon0 4394  ωcom 4622   × cxp 4657  ^◡ccnv 4658   ∘ ccom 4663   Fn wfn 5249  ⟶wf 5250  –onto→wfo 5252  –1-1-onto→wf1o 5253  ‘cfv 5254  (class class class)co 5918  freccfrec 6443  1_oc1o 6462  2_oc2o 6463  infcinf 7042  ℕ_∞xnninf 7178  Omnicomni 7193  ℝcr 7871  0cc0 7872  1c1 7873   + caddc 7875  +∞cpnf 8051   < clt 8054  ℕ₀cn0 9240  ℕ₀^*cxnn0 9303  ℤcz 9317  ℤ_≥cuz 9592  ...cfz 10074

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-map 6704  df-sup 7043  df-inf 7044  df-nninf 7179  df-omni 7194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-xnn0 9304  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209

This theorem is referenced by:  nninfct  12178