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Theorem upgrwlkdvdelem 30256

Description: Lemma for upgrwlkdvde 30257. (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 14645	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin)
2		wrdf 14631	. . 3 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
3		simpr 490	. . . . . . . . 9 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
4	3	adantr 486	. . . . . . . 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 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑥)))
6		fveq2 6873	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘𝑘) = (𝑃‘𝑥))
7		fvoveq1 7431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
8	6, 7	preq12d 4701	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})
9	5, 8	eqeq12d 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑥 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
10	9	rspcv 3572	. . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑦)))
12		fveq2 6873	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘𝑘) = (𝑃‘𝑦))
13		fvoveq1 7431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
14	12, 13	preq12d 4701	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑦 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
15	11, 14	eqeq12d 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑦 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))}))
16	15	rspcv 3572	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑦)))
19		simpl 488	. . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . 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 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‘𝑦)) = {(𝑃‘𝑦), (𝑃‘(𝑦 + 1))})
25	21, 22, 24	3eqtrd 2799	. . . . . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑥) ∈ V
27		fvex 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑥 + 1)) ∈ V
28		fvex 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘𝑦) ∈ V
29		fvex 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃‘(𝑦 + 1)) ∈ V
30	26, 27, 28, 29	preq12b 4809	. . . . . . . . . . . . . . . . . . 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 7246	. . . . . . . . . . . . . . . . . . . . 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 13779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹)))
33		elfzofz 13779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ (0...(♯‘𝐹)))
34		fveqeq2 6882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑥 → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑏)))
35		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑦 → (𝑃‘𝑏) = (𝑃‘𝑦))
38	37	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑦 → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘𝑦)))
39		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . 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 14468	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
47	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹))))
48		fzofzp1 13868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → (𝑦 + 1) ∈ (0...(♯‘𝐹)))
49	47, 48	anim12d1 622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹)))))
50	49	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))))
51		fveq2 6873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑏 = (𝑦 + 1) → (𝑃‘𝑏) = (𝑃‘(𝑦 + 1)))
52	51	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑦 + 1) → ((𝑃‘𝑥) = (𝑃‘𝑏) ↔ (𝑃‘𝑥) = (𝑃‘(𝑦 + 1))))
53		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹)))))
61	60	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))))
62		fveqeq2 6882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = (𝑥 + 1) → ((𝑃‘𝑎) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑏)))
63		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑦 → ((𝑃‘(𝑥 + 1)) = (𝑃‘𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃‘𝑦)))
66		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7415	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
74	73	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
75	74	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
76		elfzonn0 13811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℕ₀)
77		nn0cn 12586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
78		add1p1 12567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) + 1) = (𝑦 + 2))
79	77, 78	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 1) + 1) = (𝑦 + 2))
80	79	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1) + 1) = 𝑦 ↔ (𝑦 + 2) = 𝑦))
81		2cnd 12391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
82		2ne0 12419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 2 ≠ 0
83	82	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ₀ → 2 ≠ 0)
84		addn0nid 11706	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑦 + 2) ≠ 𝑦)
85	77, 81, 83, 84	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 2) ≠ 𝑦)
86		eqneqall 2966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝑦 + 1) + 1) = 𝑦 → 𝑥 = 𝑦))
91	90	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
94	93	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . . . . . . . . . . . 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 871	. . . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . 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 413	. . . . . . . . . 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 3203	. . . . . . . . 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 728	. . . . . . . 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 7246	. . . . . . . 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 595	. . . . . . 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 6532	. . . . . . 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 490	. . . . . 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 418	. . . 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 421	. . 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 596	. 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 413	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 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 {cpr 4585 ^◡ccnv 5646 dom cdm 5647 Fun wfun 6521 ⟶wf 6523 –1-1→wf1 6524 ‘cfv 6527 (class class class)co 7408 Fincfn 8951 ℂcc 11170 0cc0 11172 1c1 11173 + caddc 11175 2c2 12367 ℕ₀cn0 12576 ...cfz 13609 ..^cfzo 13757 ♯chash 14442 Word cword 14626

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-n0 12577 df-z 12664 df-uz 12936 df-fz 13610 df-fzo 13758 df-hash 14443 df-word 14627

This theorem is used by: upgrwlkdvde 30257

W3C validator