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Theorem ennnfonelemex 13305

Description: Lemma for ennnfone 13316. 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 4547	. . . . 5 ⊢ (𝑛 = (^◡𝑁‘𝑃) → suc 𝑛 = suc (^◡𝑁‘𝑃))
2	1	raleqdv 2755	. . . 4 ⊢ (𝑛 = (^◡𝑁‘𝑃) → (∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗)))
3	2	rexbidv 2551	. . 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 10865	. . . . . 6 ⊢ 𝑁:ω–1-1-onto→ℕ₀
7		f1ocnv 5652	. . . . . 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 5639	. . . . 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 5844	. . 3 ⊢ (𝜑 → (^◡𝑁‘𝑃) ∈ ω)
13	3, 4, 12	rspcdva 2934	. 2 ⊢ (𝜑 → ∃𝑘 ∈ ω ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
14		f1of 5639	. . . . 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 4742	. . . . 5 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
17	16	ad2antrl 494	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → suc 𝑘 ∈ ω)
18	15, 17	ffvelcdmd 5844	. . 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 492	. . . . . . . 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 5616	. . . . . . . 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 2824	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
24	23	sucid 4562	. . . . . . . . 9 ⊢ 𝑘 ∈ suc 𝑘
25		simprl 535	. . . . . . . . . . . 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 5615	. . . . . . . . . . . 12 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
28		fdm 5539	. . . . . . . . . . . 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 2318	. . . . . . . . . 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 5950	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑘 ∈ dom 𝐹) → (𝑘 ∈ suc 𝑘 → (𝐹‘𝑘) ∈ (𝐹 “ suc 𝑘)))
32	22, 30, 31	syl2anc 415	. . . . . . . . 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 5695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑎 → (𝐹‘𝑗) = (𝐹‘𝑎))
40	39	neeq2d 2439	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑎 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑎)))
41	40	cbvralv 2786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎))
42	41	rexbii 2557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ω ∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎))
43		fveq2 5695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
44	43	neeq1d 2438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) ≠ (𝐹‘𝑎) ↔ (𝐹‘𝑏) ≠ (𝐹‘𝑎)))
45	44	ralbidv 2550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑏 → (∀𝑎 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑎) ↔ ∀𝑎 ∈ suc 𝑛(𝐹‘𝑏) ≠ (𝐹‘𝑎)))
46	45	cbvrexv 2787	. . . . . . . . . . . . . . . . . . . 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 2556	. . . . . . . . . . . . . . . . . 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 13304	. . . . . . . . . . . . . . . 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 5641	. . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . 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 5844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝑁‘suc 𝑘) ∈ ℕ₀)
61	60	adantrr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘suc 𝑘) ∈ ℕ₀)
62	57	nn0red 9621	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑃 ∈ ℝ)
63	61	nn0red 9621	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝑁‘suc 𝑘) ∈ ℝ)
64		f1ocnvfv2 5984	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁:ω–1-1-onto→ℕ₀ ∧ 𝑃 ∈ ℕ₀) → (𝑁‘(^◡𝑁‘𝑃)) = 𝑃)
65	6, 57, 64	sylancr 418	. . . . . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . . . . . . . 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 13291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (^◡𝑁‘𝑃) ∈ 𝑘)
69		elelsuc 4554	. . . . . . . . . . . . . . . . . . . 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 9656	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 0 ∈ ℤ)
72	71, 5, 66, 17	frec2uzltd 10840	. . . . . . . . . . . . . . . . . . 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 4154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → 𝑃 < (𝑁‘suc 𝑘))
75	62, 63, 74	ltled 8445	. . . . . . . . . . . . . . . 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 13302	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐻‘𝑃) ⊆ (𝐻‘(𝑁‘suc 𝑘)))
77	76	ad2antrr 492	. . . . . . . . . . . . . 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 5721	. . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . 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 2244	. . . . . . . . . . . 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 2623	. . . . . . . . . . 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 13304	. . . . . . . . . . . . . 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 5641	. . . . . . . . . . . . . 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 5811	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 957	. . . . . . . . . 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 5011	. . . . . . . . 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 5648	. . . . . . . . . . . 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 5983	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁:ω–1-1-onto→ℕ₀ ∧ suc 𝑘 ∈ ω) → (^◡𝑁‘(𝑁‘suc 𝑘)) = suc 𝑘)
96	6, 17, 95	sylancr 418	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (^◡𝑁‘(𝑁‘suc 𝑘)) = suc 𝑘)
97	96	imaeq2d 5126	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → (𝐹 “ (^◡𝑁‘(𝑁‘suc 𝑘))) = (𝐹 “ suc 𝑘))
98		f1oeq3 5629	. . . . . . . . . . . . . 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 5648	. . . . . . . . . . . 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 2279	. . . . . . . 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 2318	. . . . . . 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 5754	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝑘) ∈ (𝐹 “ (^◡𝑁‘𝑃))) → ∃𝑞 ∈ (^◡𝑁‘𝑃)(𝐹‘𝑞) = (𝐹‘𝑘))
108	22, 106, 107	syl2anc 415	. . . . . 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 537	. . . . . . 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 5695	. . . . . . . . . 10 ⊢ (𝑗 = 𝑞 → (𝐹‘𝑗) = (𝐹‘𝑞))
111	110	neeq2d 2439	. . . . . . . . 9 ⊢ (𝑗 = 𝑞 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑞)))
112	67	ad2antrr 492	. . . . . . . . 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 4554	. . . . . . . . . 10 ⊢ (𝑞 ∈ (^◡𝑁‘𝑃) → 𝑞 ∈ suc (^◡𝑁‘𝑃))
114	113	ad2antrl 494	. . . . . . . . 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 2934	. . . . . . . 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 2506	. . . . . . 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 2503	. . . . . 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 2673	. . . . 5 ⊢ (((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) ∧ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘))) → ⊥)
119	118	inegd 1421	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ¬ dom (𝐻‘𝑃) = dom (𝐻‘(𝑁‘suc 𝑘)))
120		dmss 4980	. . . . . 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 13299	. . . . . . 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 2550	. . . . . . . 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 13299	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → dom (𝐻‘(𝑁‘suc 𝑘)) ∈ ω)
128		nntri1 6769	. . . . . 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 415	. . . . 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 6766	. . . . 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 415	. . . 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 1391	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘)))
134		fveq2 5695	. . . . . 6 ⊢ (𝑖 = (𝑁‘suc 𝑘) → (𝐻‘𝑖) = (𝐻‘(𝑁‘suc 𝑘)))
135	134	dmeqd 4983	. . . . 5 ⊢ (𝑖 = (𝑁‘suc 𝑘) → dom (𝐻‘𝑖) = dom (𝐻‘(𝑁‘suc 𝑘)))
136	135	eleq2d 2308	. . . 4 ⊢ (𝑖 = (𝑁‘suc 𝑘) → (dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖) ↔ dom (𝐻‘𝑃) ∈ dom (𝐻‘(𝑁‘suc 𝑘))))
137	136	rspcev 2929	. . 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 415	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ ∀𝑗 ∈ suc (^◡𝑁‘𝑃)(𝐹‘𝑘) ≠ (𝐹‘𝑗))) → ∃𝑖 ∈ ℕ₀ dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))
139	13, 138	rexlimddv 2673	1 ⊢ (𝜑 → ∃𝑖 ∈ ℕ₀ dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 846 ∨ w3o 1008 = wceq 1402 ⊥wfal 1407 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 ∃wrex 2529 ∪ cun 3218 ⊆ wss 3220 ∅c0 3520 ifcif 3638 {csn 3709 ⟨cop 3712 class class class wbr 4130 ↦ cmpt 4192 suc csuc 4510 ωcom 4737 ^◡ccnv 4773 dom cdm 4774 ran crn 4775 “ cima 4777 Fun wfun 5371 ⟶wf 5373 –1-1→wf1 5374 –onto→wfo 5375 –1-1-onto→wf1o 5376 ‘cfv 5377 (class class class)co 6085 ∈ cmpo 6087 freccfrec 6661 ↑_pm cpm 6923 0cc0 8179 1c1 8180 + caddc 8182 < clt 8360 − cmin 8497 ℕ₀cn0 9563 ℤcz 9644 seqcseq 10884

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pm 6925 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-n0 9564 df-z 9645 df-uz 9922 df-seqfrec 10885

This theorem is used by: ennnfonelemhom 13306

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