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Theorem upgrwlkdvdelem 30096

Description: Lemma for upgrwlkdvde 30097. (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 14576	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin)
2		wrdf 14562	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
3		simpr 489	. . . . . . . . 9 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
4	3	adantr 485	. . . . . . . 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 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑥)))
6		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘𝑘) = (𝑃‘𝑥))
7		fvoveq1 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
8	6, 7	preq12d 4706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})
9	5, 8	eqeq12d 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
10	9	rspcv 3576	. . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑦)))
12		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘𝑘) = (𝑃‘𝑦))
13		fvoveq1 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
14	12, 13	preq12d 4706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
15	11, 14	eqeq12d 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}))
16	15	rspcv 3576	. . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)))
19		simpl 487	. . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑥) ∈ V
27		fvex 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑥 + 1)) ∈ V
28		fvex 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑦) ∈ V
29		fvex 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑦 + 1)) ∈ V
30	26, 27, 28, 29	preq12b 4814	. . . . . . . . . . . . . . . . . . 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 7252	. . . . . . . . . . . . . . . . . . . . 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 13711	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹)))
33		elfzofz 13711	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ (0...(♯‘𝐹)))
34		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑏)))
35		eqeq1 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → (𝑎 = 𝑏 ↔ 𝑥 = 𝑏))
36	34, 35	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑥 → (((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏)))
37		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑦 → (𝑃‘𝑏) = (𝑃‘𝑦))
38	37	eqeq2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑦)))
39		eqeq2 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → (𝑥 = 𝑏 ↔ 𝑥 = 𝑦))
40	38, 39	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏 = 𝑦 → (((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘𝑦) → 𝑥 = 𝑦)))
41	36, 40	rspc2v 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 607	. . . . . . . . . . . . . . . . . . . . . . . . . 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 51	. . . . . . . . . . . . . . . . . . . . . . . . 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 97	. . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . 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 14399	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
47	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹))))
48		fzofzp1 13800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (𝑦 + 1) ∈ (0...(♯‘𝐹)))
49	47, 48	anim12d1 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹)))))
50	49	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))))
51		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑏 = (𝑦 + 1) → (𝑃‘𝑏) = (𝑃‘(𝑦 + 1)))
52	51	eqeq2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘(𝑦 + 1))))
53		eqeq2 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → (𝑥 = 𝑏 ↔ 𝑥 = (𝑦 + 1)))
54	52, 53	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = (𝑦 + 1) → (((𝑃‘𝑥) = (𝑃‘𝑏) → 𝑥 = 𝑏) ↔ ((𝑃‘𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
55	36, 54	rspc2v 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹)))))
61	60	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))))
62		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑏)))
63		eqeq1 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → (𝑎 = 𝑏 ↔ (𝑥 + 1) = 𝑏))
64	62, 63	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑎 = (𝑥 + 1) → (((𝑃‘𝑎) = (𝑃‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) → (𝑥 + 1) = 𝑏)))
65	37	eqeq2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)))
66		eqeq2 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑥 + 1) = 𝑏 ↔ (𝑥 + 1) = 𝑦))
67	65, 66	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = 𝑦 → (((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) → (𝑥 + 1) = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑦) → (𝑥 + 1) = 𝑦)))
68	64, 67	rspc2v 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
74	73	eqeq1d 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
75	74	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
76		elfzonn0 13743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℕ₀)
77		nn0cn 12520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
78		add1p1 12501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) + 1) = (𝑦 + 2))
79	77, 78	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 1) + 1) = (𝑦 + 2))
80	79	eqeq1d 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1) + 1) = 𝑦 ↔ (𝑦 + 2) = 𝑦))
81		2cnd 12325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
82		2ne0 12353	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 2 ≠ 0
83	82	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ≠ 0)
84		addn0nid 11640	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑦 + 2) ≠ 𝑦)
85	77, 81, 83, 84	syl3anc 1397	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 2) ≠ 𝑦)
86		eqneqall 2968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 + 2) = 𝑦 → ((𝑦 + 2) ≠ 𝑦 → 𝑥 = 𝑦))
87	85, 86	syl5com 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 2) = 𝑦 → 𝑥 = 𝑦))
88	80, 87	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
89	76, 88	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
90	89	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
91	90	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
92	75, 91	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 → 𝑥 = 𝑦))
93	92	expimpd 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
94	93	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
95	72, 94	syld 48	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 90	. . . . . . . . . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . . . . . . . . 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 92	. . . . . . . . . . . . . . . . . . . . . . . 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 97	. . . . . . . . . . . . . . . . . . . . . . 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 870	. . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 102	. . . . . . . . . . . . . 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 48	. . . . . . . . . . . . 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 97	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 412	. . . . . . . . . 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 3205	. . . . . . . . 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 727	. . . . . . . 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 7252	. . . . . . . 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 594	. . . . . . 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 6541	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ^◡𝐹))
120	118, 119	sylib 221	. . . . . 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 489	. . . . . 6 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ^◡𝐹) → Fun ^◡𝐹)
122	120, 121	syl 18	. . . . 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 417	. . . 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 420	. . 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 595	. 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 412	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 {cpr 4590 ^◡ccnv 5659 dom cdm 5660 Fun wfun 6530 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 2c2 12301 ℕ₀cn0 12510 ...cfz 13541 ..^cfzo 13689 ♯chash 14373 Word cword 14557

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-n0 12511 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-hash 14374 df-word 14558

This theorem is used by: upgrwlkdvde 30097

W3C validator