ennnfonelemex - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ennnfonelemex		GIF version

Theorem ennnfonelemex 12985

Description: Lemma for ennnfone 12996. Extending the sequence (𝐻‘𝑃) to include an additional element. (Contributed by Jim Kingdon, 19-Jul-2023.)

**Hypotheses**
Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)
ennnfonelemex.p	⊢ (𝜑 → 𝑃 ∈ ℕ₀)

**Assertion**
Ref	Expression
ennnfonelemex	⊢ (𝜑 → ∃𝑖 ∈ ℕ₀ dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))

Distinct variable groups: 𝐴,𝑗,𝑥,𝑦 𝑗,𝐹,𝑘,𝑛 𝑥,𝐹,𝑦 𝑗,𝐺 𝑗,𝐻,𝑘,𝑛 𝑖,𝐻,𝑘 𝑥,𝐻,𝑦,𝑘 𝑗,𝐽 𝑗,𝑁,𝑘,𝑛 𝑖,𝑁 𝑥,𝑁,𝑦 𝑃,𝑗,𝑘,𝑛 𝑥,𝑃,𝑦 𝑃,𝑖 𝜑,𝑗,𝑘,𝑛 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑖) 𝐴(𝑖,𝑘,𝑛) 𝐹(𝑖) 𝐺(𝑥,𝑦,𝑖,𝑘,𝑛) 𝐽(𝑥,𝑦,𝑖,𝑘,𝑛)

Proof of Theorem ennnfonelemex Dummy variables 𝑎 𝑏 𝑞 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 4493	. . . . 5 ⊢ (𝑛 = (^◡𝑁‘𝑃) → suc 𝑛 = suc (^◡𝑁‘𝑃))
2	1	raleqdv 2734	. . . 4 ⊢ (𝑛 = (^◡𝑁‘𝑃) → (∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
3	2	rexbidv 2531	. . 3 ⊢ (𝑛 = (^◡𝑁‘𝑃) → (∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ω ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
4		ennnfonelemh.ne	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
5		ennnfonelemh.n	. . . . . . 7 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
6	5	frechashgf1o 10650	. . . . . 6 ⊢ 𝑁:ω–1-1-onto→ℕ₀
7		f1ocnv 5585	. . . . . 6 ⊢ (𝑁:ω–1-1-onto→ℕ₀ → ^◡𝑁:ℕ₀–1-1-onto→ω)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ^◡𝑁:ℕ₀–1-1-onto→ω
9		f1of 5572	. . . . 5 ⊢ (^◡𝑁:ℕ₀–1-1-onto→ω → ^◡𝑁:ℕ₀⟶ω)
10	8, 9	mp1i 10	. . . 4 ⊢ (𝜑 → ^◡𝑁:ℕ₀⟶ω)
11		ennnfonelemex.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
12	10, 11	ffvelcdmd 5771	. . 3 ⊢ (𝜑 → (^◡𝑁‘𝑃) ∈ ω)
13	3, 4, 12	rspcdva 2912	. 2 ⊢ (𝜑 → ∃𝑘 ∈ ω ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
14		f1of 5572	. . . . 5 ⊢ (𝑁:ω–1-1-onto→ℕ₀ → 𝑁:ω⟶ℕ₀)
15	6, 14	mp1i 10	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑁:ω⟶ℕ₀)
16		peano2 4687	. . . . 5 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
17	16	ad2antrl 490	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → suc 𝑘 ∈ ω)
18	15, 17	ffvelcdmd 5771	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘suc 𝑘) ∈ ℕ₀)
19		ennnfonelemh.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
20	19	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → 𝐹:ω–onto→𝐴)
21		fofun 5549	. . . . . . . 8 ⊢ (𝐹:ω–onto→𝐴 → Fun 𝐹)
22	20, 21	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → Fun 𝐹)
23		vex 2802	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
24	23	sucid 4508	. . . . . . . . 9 ⊢ 𝑘 ∈ suc 𝑘
25		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑘 ∈ ω)
26	25	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → 𝑘 ∈ ω)
27		fof 5548	. . . . . . . . . . . 12 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
28		fdm 5479	. . . . . . . . . . . 12 ⊢ (𝐹:ω⟶𝐴 → dom 𝐹 = ω)
29	20, 27, 28	3syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → dom 𝐹 = ω)
30	26, 29	eleqtrrd 2309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → 𝑘 ∈ dom 𝐹)
31		funfvima 5871	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑘 ∈ dom 𝐹) → (𝑘 ∈ suc 𝑘 → (𝐹‘𝑘) ∈ (𝐹 “ suc 𝑘)))
32	22, 30, 31	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → (𝑘 ∈ suc 𝑘 → (𝐹‘𝑘) ∈ (𝐹 “ suc 𝑘)))
33	24, 32	mpi 15	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → (𝐹‘𝑘) ∈ (𝐹 “ suc 𝑘))
34		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)))
35		ennnfonelemh.dceq	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
36	35	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
37	19	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝐹:ω–onto→𝐴)
38	4	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
39		fveq2 5627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
40	39	neeq2d 2419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑎)))
41	40	cbvralv 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎))
42	41	rexbii 2537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎))
43		fveq2 5627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
44	43	neeq1d 2418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) ≠ (𝐹‘𝑎) ↔ (𝐹‘𝑏) ≠ (𝐹‘𝑎)))
45	44	ralbidv 2530	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑏 → (∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎) ↔ ∀𝑎 ∈ suc 𝑛(𝐹‘𝑏) ≠ (𝐹‘𝑎)))
46	45	cbvrexv 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑘 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎) ↔ ∃𝑏 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑏) ≠ (𝐹‘𝑎))
47	42, 46	bitri 184	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑏 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑏) ≠ (𝐹‘𝑎))
48	47	ralbii 2536	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑛 ∈ ω ∃𝑏 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑏) ≠ (𝐹‘𝑎))
49	38, 48	sylib 122	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∀𝑛 ∈ ω ∃𝑏 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑏) ≠ (𝐹‘𝑎))
50		ennnfonelemh.g	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
51		ennnfonelemh.j	. . . . . . . . . . . . . . . . 17 ⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
52		ennnfonelemh.h	. . . . . . . . . . . . . . . . 17 ⊢ 𝐻 = seq0(𝐺, 𝐽)
53	36, 37, 49, 50, 5, 51, 52, 18	ennnfonelemhf1o 12984	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))))
54		f1ofun 5574	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))) → Fun (𝐻‘(𝑁‘suc 𝑘)))
55	53, 54	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → Fun (𝐻‘(𝑁‘suc 𝑘)))
56	55	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ 𝑠 ∈ dom (𝐻‘𝑃)) → Fun (𝐻‘(𝑁‘suc 𝑘)))
57	11	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑃 ∈ ℕ₀)
58	6, 14	mp1i 10	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → 𝑁:ω⟶ℕ₀)
59	16	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
60	58, 59	ffvelcdmd 5771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝑁‘suc 𝑘) ∈ ℕ₀)
61	60	adantrr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘suc 𝑘) ∈ ℕ₀)
62	57	nn0red 9423	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑃 ∈ ℝ)
63	61	nn0red 9423	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘suc 𝑘) ∈ ℝ)
64		f1ocnvfv2 5902	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁:ω–1-1-onto→ℕ₀ ∧ 𝑃 ∈ ℕ₀) → (𝑁‘(^◡𝑁‘𝑃)) = 𝑃)
65	6, 57, 64	sylancr 414	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘(^◡𝑁‘𝑃)) = 𝑃)
66	12	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (^◡𝑁‘𝑃) ∈ ω)
67		simprr 531	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
68	37, 25, 66, 67	ennnfonelemk 12971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (^◡𝑁‘𝑃) ∈ 𝑘)
69		elelsuc 4500	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((^◡𝑁‘𝑃) ∈ 𝑘 → (^◡𝑁‘𝑃) ∈ suc 𝑘)
70	68, 69	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (^◡𝑁‘𝑃) ∈ suc 𝑘)
71		0zd 9458	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 0 ∈ ℤ)
72	71, 5, 66, 17	frec2uzltd 10625	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ((^◡𝑁‘𝑃) ∈ suc 𝑘 → (𝑁‘(^◡𝑁‘𝑃)) < (𝑁‘suc 𝑘)))
73	70, 72	mpd 13	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘(^◡𝑁‘𝑃)) < (𝑁‘suc 𝑘))
74	65, 73	eqbrtrrd 4107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑃 < (𝑁‘suc 𝑘))
75	62, 63, 74	ltled 8265	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑃 ≤ (𝑁‘suc 𝑘))
76	36, 37, 38, 50, 5, 51, 52, 57, 61, 75	ennnfoneleminc 12982	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐻‘𝑃) ⊆ (𝐻‘(𝑁‘suc 𝑘)))
77	76	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ 𝑠 ∈ dom (𝐻‘𝑃)) → (𝐻‘𝑃) ⊆ (𝐻‘(𝑁‘suc 𝑘)))
78		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ 𝑠 ∈ dom (𝐻‘𝑃)) → 𝑠 ∈ dom (𝐻‘𝑃))
79		funssfv 5653	. . . . . . . . . . . . . 14 ⊢ ((Fun (𝐻‘(𝑁‘suc 𝑘)) ∧ (𝐻‘𝑃) ⊆ (𝐻‘(𝑁‘suc 𝑘)) ∧ 𝑠 ∈ dom (𝐻‘𝑃)) → ((𝐻‘(𝑁‘suc 𝑘))‘𝑠) = ((𝐻‘𝑃)‘𝑠))
80	56, 77, 78, 79	syl3anc 1271	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ 𝑠 ∈ dom (𝐻‘𝑃)) → ((𝐻‘(𝑁‘suc 𝑘))‘𝑠) = ((𝐻‘𝑃)‘𝑠))
81	80	eqcomd 2235	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ 𝑠 ∈ dom (𝐻‘𝑃)) → ((𝐻‘𝑃)‘𝑠) = ((𝐻‘(𝑁‘suc 𝑘))‘𝑠))
82	81	ralrimiva 2603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ∀𝑠 ∈ dom (𝐻‘𝑃)((𝐻‘𝑃)‘𝑠) = ((𝐻‘(𝑁‘suc 𝑘))‘𝑠))
83	36, 37, 49, 50, 5, 51, 52, 57	ennnfonelemhf1o 12984	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐻‘𝑃):dom (𝐻‘𝑃)–1-1-onto→(𝐹 “ (^◡𝑁‘𝑃)))
84		f1ofun 5574	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑃):dom (𝐻‘𝑃)–1-1-onto→(𝐹 “ (^◡𝑁‘𝑃)) → Fun (𝐻‘𝑃))
85	83, 84	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → Fun (𝐻‘𝑃))
86		eqfunfv 5737	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐻‘𝑃) ∧ Fun (𝐻‘(𝑁‘suc 𝑘))) → ((𝐻‘𝑃) = (𝐻‘(𝑁‘suc 𝑘)) ↔ (dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)) ∧ ∀𝑠 ∈ dom (𝐻‘𝑃)((𝐻‘𝑃)‘𝑠) = ((𝐻‘(𝑁‘suc 𝑘))‘𝑠))))
87	85, 55, 86	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ((𝐻‘𝑃) = (𝐻‘(𝑁‘suc 𝑘)) ↔ (dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)) ∧ ∀𝑠 ∈ dom (𝐻‘𝑃)((𝐻‘𝑃)‘𝑠) = ((𝐻‘(𝑁‘suc 𝑘))‘𝑠))))
88	87	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ((𝐻‘𝑃) = (𝐻‘(𝑁‘suc 𝑘)) ↔ (dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)) ∧ ∀𝑠 ∈ dom (𝐻‘𝑃)((𝐻‘𝑃)‘𝑠) = ((𝐻‘(𝑁‘suc 𝑘))‘𝑠))))
89	34, 82, 88	mpbir2and 950	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → (𝐻‘𝑃) = (𝐻‘(𝑁‘suc 𝑘)))
90	89	rneqd 4953	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ran (𝐻‘𝑃) = ran (𝐻‘(𝑁‘suc 𝑘)))
91		dff1o5 5581	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑃):dom (𝐻‘𝑃)–1-1-onto→(𝐹 “ (^◡𝑁‘𝑃)) ↔ ((𝐻‘𝑃):dom (𝐻‘𝑃)–1-1→(𝐹 “ (^◡𝑁‘𝑃)) ∧ ran (𝐻‘𝑃) = (𝐹 “ (^◡𝑁‘𝑃))))
92	83, 91	sylib 122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ((𝐻‘𝑃):dom (𝐻‘𝑃)–1-1→(𝐹 “ (^◡𝑁‘𝑃)) ∧ ran (𝐻‘𝑃) = (𝐹 “ (^◡𝑁‘𝑃))))
93	92	simprd 114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ran (𝐻‘𝑃) = (𝐹 “ (^◡𝑁‘𝑃)))
94	93	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ran (𝐻‘𝑃) = (𝐹 “ (^◡𝑁‘𝑃)))
95		f1ocnvfv1 5901	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁:ω–1-1-onto→ℕ₀ ∧ suc 𝑘 ∈ ω) → (^◡𝑁‘(𝑁‘suc 𝑘)) = suc 𝑘)
96	6, 17, 95	sylancr 414	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (^◡𝑁‘(𝑁‘suc 𝑘)) = suc 𝑘)
97	96	imaeq2d 5068	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))) = (𝐹 “ suc 𝑘))
98		f1oeq3 5562	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))) = (𝐹 “ suc 𝑘) → ((𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))) ↔ (𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ suc 𝑘)))
99	97, 98	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ((𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))) ↔ (𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ suc 𝑘)))
100	53, 99	mpbid 147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ suc 𝑘))
101		dff1o5 5581	. . . . . . . . . . . 12 ⊢ ((𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1-onto→(𝐹 “ suc 𝑘) ↔ ((𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1→(𝐹 “ suc 𝑘) ∧ ran (𝐻‘(𝑁‘suc 𝑘)) = (𝐹 “ suc 𝑘)))
102	100, 101	sylib 122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ((𝐻‘(𝑁‘suc 𝑘)):dom (𝐻‘(𝑁‘suc 𝑘))–1-1→(𝐹 “ suc 𝑘) ∧ ran (𝐻‘(𝑁‘suc 𝑘)) = (𝐹 “ suc 𝑘)))
103	102	simprd 114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ran (𝐻‘(𝑁‘suc 𝑘)) = (𝐹 “ suc 𝑘))
104	103	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ran (𝐻‘(𝑁‘suc 𝑘)) = (𝐹 “ suc 𝑘))
105	90, 94, 104	3eqtr3d 2270	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → (𝐹 “ (^◡𝑁‘𝑃)) = (𝐹 “ suc 𝑘))
106	33, 105	eleqtrrd 2309	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → (𝐹‘𝑘) ∈ (𝐹 “ (^◡𝑁‘𝑃)))
107		fvelima 5685	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝑘) ∈ (𝐹 “ (^◡𝑁‘𝑃))) → ∃𝑞 ∈ (^◡𝑁‘𝑃)(𝐹‘𝑞) = (𝐹‘𝑘))
108	22, 106, 107	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ∃𝑞 ∈ (^◡𝑁‘𝑃)(𝐹‘𝑞) = (𝐹‘𝑘))
109		simprr 531	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ (𝑞 ∈ (^◡𝑁‘𝑃) ∧ (𝐹‘𝑞) = (𝐹‘𝑘))) → (𝐹‘𝑞) = (𝐹‘𝑘))
110		fveq2 5627	. . . . . . . . . 10 ⊢ (𝑗 = 𝑞 → (𝐹‘𝑗) = (𝐹‘𝑞))
111	110	neeq2d 2419	. . . . . . . . 9 ⊢ (𝑗 = 𝑞 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑞)))
112	67	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ (𝑞 ∈ (^◡𝑁‘𝑃) ∧ (𝐹‘𝑞) = (𝐹‘𝑘))) → ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
113		elelsuc 4500	. . . . . . . . . 10 ⊢ (𝑞 ∈ (^◡𝑁‘𝑃) → 𝑞 ∈ suc (^◡𝑁‘𝑃))
114	113	ad2antrl 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ (𝑞 ∈ (^◡𝑁‘𝑃) ∧ (𝐹‘𝑞) = (𝐹‘𝑘))) → 𝑞 ∈ suc (^◡𝑁‘𝑃))
115	111, 112, 114	rspcdva 2912	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ (𝑞 ∈ (^◡𝑁‘𝑃) ∧ (𝐹‘𝑞) = (𝐹‘𝑘))) → (𝐹‘𝑘) ≠ (𝐹‘𝑞))
116	115	necomd 2486	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ (𝑞 ∈ (^◡𝑁‘𝑃) ∧ (𝐹‘𝑞) = (𝐹‘𝑘))) → (𝐹‘𝑞) ≠ (𝐹‘𝑘))
117	109, 116	pm2.21ddne 2483	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) ∧ (𝑞 ∈ (^◡𝑁‘𝑃) ∧ (𝐹‘𝑞) = (𝐹‘𝑘))) → ⊥)
118	108, 117	rexlimddv 2653	. . . . 5 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ⊥)
119	118	inegd 1414	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ¬ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)))
120		dmss 4922	. . . . . 6 ⊢ ((𝐻‘𝑃) ⊆ (𝐻‘(𝑁‘suc 𝑘)) → dom (𝐻‘𝑃) ⊆ dom (𝐻‘(𝑁‘suc 𝑘)))
121	76, 120	syl 14	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → dom (𝐻‘𝑃) ⊆ dom (𝐻‘(𝑁‘suc 𝑘)))
122	35, 19, 4, 50, 5, 51, 52, 11	ennnfonelemom 12979	. . . . . . 7 ⊢ (𝜑 → dom (𝐻‘𝑃) ∈ ω)
123	122	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → dom (𝐻‘𝑃) ∈ ω)
124	42	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎)))
125	124	ralbidv 2530	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎)))
126	38, 125	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎))
127	36, 37, 126, 50, 5, 51, 52, 61	ennnfonelemom 12979	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → dom (𝐻‘(𝑁‘suc 𝑘)) ∈ ω)
128		nntri1 6642	. . . . . 6 ⊢ ((dom (𝐻‘𝑃) ∈ ω ∧ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ ω) → (dom (𝐻‘𝑃) ⊆ dom (𝐻‘(𝑁‘suc 𝑘)) ↔ ¬ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ dom (𝐻‘𝑃)))
129	123, 127, 128	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (dom (𝐻‘𝑃) ⊆ dom (𝐻‘(𝑁‘suc 𝑘)) ↔ ¬ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ dom (𝐻‘𝑃)))
130	121, 129	mpbid 147	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ¬ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ dom (𝐻‘𝑃))
131		nntri3or 6639	. . . . 5 ⊢ ((dom (𝐻‘𝑃) ∈ ω ∧ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ ω) → (dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘)) ∨ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)) ∨ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ dom (𝐻‘𝑃)))
132	123, 127, 131	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘)) ∨ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)) ∨ dom (𝐻‘(𝑁‘suc 𝑘)) ∈ dom (𝐻‘𝑃)))
133	119, 130, 132	ecase23d 1384	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘)))
134		fveq2 5627	. . . . . 6 ⊢ (𝑖 = (𝑁‘suc 𝑘) → (𝐻‘𝑖) = (𝐻‘(𝑁‘suc 𝑘)))
135	134	dmeqd 4925	. . . . 5 ⊢ (𝑖 = (𝑁‘suc 𝑘) → dom (𝐻‘𝑖) = dom (𝐻‘(𝑁‘suc 𝑘)))
136	135	eleq2d 2299	. . . 4 ⊢ (𝑖 = (𝑁‘suc 𝑘) → (dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖) ↔ dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘))))
137	136	rspcev 2907	. . 3 ⊢ (((𝑁‘suc 𝑘) ∈ ℕ₀ ∧ dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘))) → ∃𝑖 ∈ ℕ₀ dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))
138	18, 133, 137	syl2anc 411	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∃𝑖 ∈ ℕ₀ dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))
139	13, 138	rexlimddv 2653	1 ⊢ (𝜑 → ∃𝑖 ∈ ℕ₀ dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 839 ∨ w3o 1001 = wceq 1395 ⊥wfal 1400 ∈ wcel 2200 ≠ wne 2400 ∀wral 2508 ∃wrex 2509 ∪ cun 3195 ⊆ wss 3197 ∅c0 3491 ifcif 3602 {csn 3666 ⟨cop 3669 class class class wbr 4083 ↦ cmpt 4145 suc csuc 4456 ωcom 4682 ^◡ccnv 4718 dom cdm 4719 ran crn 4720 “ cima 4722 Fun wfun 5312 ⟶wf 5314 –1-1→wf1 5315 –onto→wfo 5316 –1-1-onto→wf1o 5317 ‘cfv 5318 (class class class)co 6001 ∈ cmpo 6003 freccfrec 6536 ↑_pm cpm 6796 0cc0 7999 1c1 8000 + caddc 8002 < clt 8181 − cmin 8317 ℕ₀cn0 9369 ℤcz 9446 seqcseq 10669

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-addcom 8099 ax-addass 8101 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-0id 8107 ax-rnegex 8108 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-ltadd 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-pm 6798 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-inn 9111 df-n0 9370 df-z 9447 df-uz 9723 df-seqfrec 10670

This theorem is referenced by: ennnfonelemhom 12986

W3C validator