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Theorem upgrwlkdvdelem 27228

Description: Lemma for upgrwlkdvde 27229. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)

**Assertion**
Ref	Expression
upgrwlkdvdelem	⊢ ((𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ 𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → Fun ^◡𝐹))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐼 𝑃,𝑘 Allowed substitution hint: 𝑉(𝑘)

Proof of Theorem upgrwlkdvdelem Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrdfin 13696	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin)
2		wrdf 13680	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
3		simpr 477	. . . . . . . . 9 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
4	3	adantr 473	. . . . . . . 8 ⊢ (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
5		2fveq3 6506	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑥)))
6		fveq2 6501	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘𝑘) = (𝑃‘𝑥))
7		fvoveq1 7001	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
8	6, 7	preq12d 4552	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})
9	5, 8	eqeq12d 2793	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
10	9	rspcv 3531	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
11		2fveq3 6506	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑦)))
12		fveq2 6501	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘𝑘) = (𝑃‘𝑦))
13		fvoveq1 7001	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
14	12, 13	preq12d 4552	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
15	11, 14	eqeq12d 2793	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}))
16	15	rspcv 3531	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}))
17	10, 16	anim12ii 608	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})))
18		fveq2 6501	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)))
19		simpl 475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}) → (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})
20	19	eqcomd 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = (𝐼‘(𝐹‘𝑥)))
21	20	adantl 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)) ∧ ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = (𝐼‘(𝐹‘𝑥)))
22		simpl 475	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)) ∧ ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)))
23		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}) → (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
24	23	adantl 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)) ∧ ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})) → (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
25	21, 22, 24	3eqtrd 2818	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)) ∧ ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
26		fvex 6514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑥) ∈ V
27		fvex 6514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑥 + 1)) ∈ V
28		fvex 6514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑦) ∈ V
29		fvex 6514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑦 + 1)) ∈ V
30	26, 27, 28, 29	preq12b 4656	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))} ↔ (((𝑃‘𝑥) = (𝑃‘𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))))
31		dff13 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ↔ (𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏)))
32		elfzofz 12872	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹)))
33		elfzofz 12872	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ (0...(♯‘𝐹)))
34		fveqeq2 6510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑏)))
35		eqeq1 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → (𝑎 = 𝑏 ↔ 𝑥 = 𝑏))
36	34, 35	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑥 → (((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏)))
37		fveq2 6501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑦 → (𝑃‘𝑏) = (𝑃‘𝑦))
38	37	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑦)))
39		eqeq2 2789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → (𝑥 = 𝑏 ↔ 𝑥 = 𝑦))
40	38, 39	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏 = 𝑦 → (((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘𝑦) → 𝑥 = 𝑦)))
41	36, 40	rspc2v 3548	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → ((𝑃‘𝑥) = (𝑃‘𝑦) → 𝑥 = 𝑦)))
42	32, 33, 41	syl2an 586	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → ((𝑃‘𝑥) = (𝑃‘𝑦) → 𝑥 = 𝑦)))
43	42	a1dd 50	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑃‘𝑥) = (𝑃‘𝑦) → 𝑥 = 𝑦))))
44	43	com14 96	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘𝑥) = (𝑃‘𝑦) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
45	44	adantr 473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃‘𝑥) = (𝑃‘𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
46		hashcl 13535	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
47	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹))))
48		fzofzp1 12952	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (𝑦 + 1) ∈ (0...(♯‘𝐹)))
49	47, 48	anim12d1 600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹)))))
50	49	imp 398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))))
51		fveq2 6501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑏 = (𝑦 + 1) → (𝑃‘𝑏) = (𝑃‘(𝑦 + 1)))
52	51	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘(𝑦 + 1))))
53		eqeq2 2789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → (𝑥 = 𝑏 ↔ 𝑥 = (𝑦 + 1)))
54	52, 53	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = (𝑦 + 1) → (((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
55	36, 54	rspc2v 3548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
56	50, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
57	56	imp 398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏)) → ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1)))
58		fzofzp1 12952	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → (𝑥 + 1) ∈ (0...(♯‘𝐹)))
59	58	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝑥 + 1) ∈ (0...(♯‘𝐹))))
60	59, 33	anim12d1 600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹)))))
61	60	imp 398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))))
62		fveqeq2 6510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑏)))
63		eqeq1 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → (𝑎 = 𝑏 ↔ (𝑥 + 1) = 𝑏))
64	62, 63	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑎 = (𝑥 + 1) → (((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) → (𝑥 + 1) = 𝑏)))
65	37	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)))
66		eqeq2 2789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑥 + 1) = 𝑏 ↔ (𝑥 + 1) = 𝑦))
67	65, 66	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = 𝑦 → (((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) → (𝑥 + 1) = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑦) → (𝑥 + 1) = 𝑦)))
68	64, 67	rspc2v 3548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑦) → (𝑥 + 1) = 𝑦)))
69	61, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑦) → (𝑥 + 1) = 𝑦)))
70	69	imp 398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏)) → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑦) → (𝑥 + 1) = 𝑦))
71	57, 70	anim12d 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏)) → (((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)) → (𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦)))
72	71	expimpd 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ∧ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → (𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦)))
73		oveq1 6985	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
74	73	eqeq1d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
75	74	adantl 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
76		elfzonn0 12900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℕ₀)
77		nn0cn 11721	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
78		add1p1 11701	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) + 1) = (𝑦 + 2))
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 1) + 1) = (𝑦 + 2))
80	79	eqeq1d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1) + 1) = 𝑦 ↔ (𝑦 + 2) = 𝑦))
81		2cnd 11521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
82		2ne0 11554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 2 ≠ 0
83	82	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ≠ 0)
84		addn0nid 10863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑦 + 2) ≠ 𝑦)
85	77, 81, 83, 84	syl3anc 1351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 2) ≠ 𝑦)
86		eqneqall 2978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 + 2) = 𝑦 → ((𝑦 + 2) ≠ 𝑦 → 𝑥 = 𝑦))
87	85, 86	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 2) = 𝑦 → 𝑥 = 𝑦))
88	80, 87	sylbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
89	76, 88	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
90	89	adantl 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
91	90	adantr 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
92	75, 91	sylbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 → 𝑥 = 𝑦))
93	92	expimpd 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
94	93	adantl 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
95	72, 94	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ∧ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → 𝑥 = 𝑦))
96	95	ex 405	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ∧ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → 𝑥 = 𝑦)))
97	46, 96	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ∧ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → 𝑥 = 𝑦)))
98	97	com3l 89	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ∧ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → (𝐹 ∈ Fin → 𝑥 = 𝑦)))
99	98	expd 408	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)) → (𝐹 ∈ Fin → 𝑥 = 𝑦))))
100	99	com34 91	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → (((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)) → 𝑥 = 𝑦))))
101	100	com14 96	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
102	45, 101	jaoi 843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑃‘𝑥) = (𝑃‘𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
103	102	adantld 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘𝑥) = (𝑃‘𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → ((𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏)) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
104	31, 103	syl5bi 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑃‘𝑥) = (𝑃‘𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
105	104	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑃‘𝑥) = (𝑃‘𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦))) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
106	30, 105	sylbi 209	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))} → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
107	25, 106	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)) ∧ ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
108	107	ex 405	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)) → (((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦)))))
109	18, 108	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → (((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦)))))
110	109	com15 101	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
111	17, 110	syld 47	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
112	111	com14 96	. . . . . . . . . . . 12 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
113	112	imp 398	. . . . . . . . . . 11 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
114	113	impcom 399	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Fin ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
115	114	ralrimivv 3140	. . . . . . . . 9 ⊢ ((𝐹 ∈ Fin ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
116	115	adantlr 702	. . . . . . . 8 ⊢ (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
117		dff13 6840	. . . . . . . 8 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
118	4, 116, 117	sylanbrc 575	. . . . . . 7 ⊢ (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
119		df-f1 6195	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ^◡𝐹))
120	118, 119	sylib 210	. . . . . 6 ⊢ (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ^◡𝐹))
121		simpr 477	. . . . . 6 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ^◡𝐹) → Fun ^◡𝐹)
122	120, 121	syl 17	. . . . 5 ⊢ (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → Fun ^◡𝐹)
123	122	ex 405	. . . 4 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → ((𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → Fun ^◡𝐹))
124	123	expd 408	. . 3 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → Fun ^◡𝐹)))
125	1, 2, 124	syl2anc 576	. 2 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝑃:(0...(♯‘𝐹))–1-1→𝑉 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → Fun ^◡𝐹)))
126	125	impcom 399	1 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ 𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → Fun ^◡𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 {cpr 4444 ^◡ccnv 5407 dom cdm 5408 Fun wfun 6184 ⟶wf 6186 –1-1→wf1 6187 ‘cfv 6190 (class class class)co 6978 Fincfn 8308 ℂcc 10335 0cc0 10337 1c1 10338 + caddc 10340 2c2 11498 ℕ₀cn0 11710 ...cfz 12711 ..^cfzo 12852 ♯chash 13508 Word cword 13675

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-fzo 12853 df-hash 13509 df-word 13676

This theorem is referenced by: upgrwlkdvde 27229

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