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Theorem upgrwlkdvdelem 29821

Description: Lemma for upgrwlkdvde 29822. (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 14467	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin)
2		wrdf 14453	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
4	3	adantr 480	. . . . . . . 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 6847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑥)))
6		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘𝑘) = (𝑃‘𝑥))
7		fvoveq1 7391	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
8	6, 7	preq12d 4700	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})
9	5, 8	eqeq12d 2753	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
10	9	rspcv 3574	. . . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑦)))
12		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘𝑘) = (𝑃‘𝑦))
13		fvoveq1 7391	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
14	12, 13	preq12d 4700	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
15	11, 14	eqeq12d 2753	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}))
16	15	rspcv 3574	. . . . . . . . . . . . . . 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 619	. . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)))
19		simpl 482	. . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑥) ∈ V
27		fvex 6855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑥 + 1)) ∈ V
28		fvex 6855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑦) ∈ V
29		fvex 6855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑦 + 1)) ∈ V
30	26, 27, 28, 29	preq12b 4808	. . . . . . . . . . . . . . . . . . 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 7210	. . . . . . . . . . . . . . . . . . . . 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 13603	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹)))
33		elfzofz 13603	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ (0...(♯‘𝐹)))
34		fveqeq2 6851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑏)))
35		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → (𝑎 = 𝑏 ↔ 𝑥 = 𝑏))
36	34, 35	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑥 → (((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏)))
37		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑦 → (𝑃‘𝑏) = (𝑃‘𝑦))
38	37	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑦)))
39		eqeq2 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → (𝑥 = 𝑏 ↔ 𝑥 = 𝑦))
40	38, 39	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏 = 𝑦 → (((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘𝑦) → 𝑥 = 𝑦)))
41	36, 40	rspc2v 3589	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 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 14291	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
47	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹))))
48		fzofzp1 13692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (𝑦 + 1) ∈ (0...(♯‘𝐹)))
49	47, 48	anim12d1 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹)))))
50	49	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))))
51		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑏 = (𝑦 + 1) → (𝑃‘𝑏) = (𝑃‘(𝑦 + 1)))
52	51	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘(𝑦 + 1))))
53		eqeq2 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → (𝑥 = 𝑏 ↔ 𝑥 = (𝑦 + 1)))
54	52, 53	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = (𝑦 + 1) → (((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
55	36, 54	rspc2v 3589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹)))))
61	60	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))))
62		fveqeq2 6851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑏)))
63		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → (𝑎 = 𝑏 ↔ (𝑥 + 1) = 𝑏))
64	62, 63	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑎 = (𝑥 + 1) → (((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) → (𝑥 + 1) = 𝑏)))
65	37	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)))
66		eqeq2 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑥 + 1) = 𝑏 ↔ (𝑥 + 1) = 𝑦))
67	65, 66	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = 𝑦 → (((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) → (𝑥 + 1) = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑦) → (𝑥 + 1) = 𝑦)))
68	64, 67	rspc2v 3589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
74	73	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
75	74	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
76		elfzonn0 13635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℕ₀)
77		nn0cn 12423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
78		add1p1 12404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) + 1) = (𝑦 + 2))
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 1) + 1) = (𝑦 + 2))
80	79	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1) + 1) = 𝑦 ↔ (𝑦 + 2) = 𝑦))
81		2cnd 12235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
82		2ne0 12261	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 2 ≠ 0
83	82	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ≠ 0)
84		addn0nid 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑦 + 2) ≠ 𝑦)
85	77, 81, 83, 84	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 2) ≠ 𝑦)
86		eqneqall 2944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
91	90	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
92	75, 91	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 → 𝑥 = 𝑦))
93	92	expimpd 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
94	93	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . . . 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 858	. . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . 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 407	. . . . . . . . . 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 3179	. . . . . . . . 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 716	. . . . . . . 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 7210	. . . . . . . 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 584	. . . . . . 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 6505	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ^◡𝐹))
120	118, 119	sylib 218	. . . . . 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 484	. . . . . 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 412	. . . 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 415	. . 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 585	. 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 407	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 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 {cpr 4584 ^◡ccnv 5631 dom cdm 5632 Fun wfun 6494 ⟶wf 6496 –1-1→wf1 6497 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 2c2 12212 ℕ₀cn0 12413 ...cfz 13435 ..^cfzo 13582 ♯chash 14265 Word cword 14448

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449

This theorem is referenced by: upgrwlkdvde 29822

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