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Theorem upgrwlkdvdelem 27528
 Description: Lemma for upgrwlkdvde 27529. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)
Assertion
Ref Expression
upgrwlkdvdelem ((𝑃:(0...(♯‘𝐹))–1-1𝑉𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (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.
StepHypRef Expression
1 wrdfin 13879 . . 3 (𝐹 ∈ Word dom 𝐼𝐹 ∈ Fin)
2 wrdf 13866 . . 3 (𝐹 ∈ Word dom 𝐼𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
3 simpr 488 . . . . . . . . 9 ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
43adantr 484 . . . . . . . 8 (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
5 2fveq3 6654 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → (𝐼‘(𝐹𝑘)) = (𝐼‘(𝐹𝑥)))
6 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑃𝑘) = (𝑃𝑥))
7 fvoveq1 7162 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
86, 7preq12d 4640 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
95, 8eqeq12d 2817 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑥 → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
109rspcv 3569 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0..^(♯‘𝐹)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
11 2fveq3 6654 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → (𝐼‘(𝐹𝑘)) = (𝐼‘(𝐹𝑦)))
12 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑃𝑘) = (𝑃𝑦))
13 fvoveq1 7162 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
1412, 13preq12d 4640 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
1511, 14eqeq12d 2817 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
1615rspcv 3569 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0..^(♯‘𝐹)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
1710, 16anim12ii 620 . . . . . . . . . . . . . 14 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})))
18 fveq2 6649 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = (𝐹𝑦) → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)))
19 simpl 486 . . . . . . . . . . . . . . . . . . . . 21 (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
2019eqcomd 2807 . . . . . . . . . . . . . . . . . . . 20 (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = (𝐼‘(𝐹𝑥)))
2120adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = (𝐼‘(𝐹𝑥)))
22 simpl 486 . . . . . . . . . . . . . . . . . . 19 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)))
23 simpr 488 . . . . . . . . . . . . . . . . . . . 20 (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
2423adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
2521, 22, 243eqtrd 2840 . . . . . . . . . . . . . . . . . 18 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
26 fvex 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑃𝑥) ∈ V
27 fvex 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑃‘(𝑥 + 1)) ∈ V
28 fvex 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑃𝑦) ∈ V
29 fvex 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑃‘(𝑦 + 1)) ∈ V
3026, 27, 28, 29preq12b 4744 . . . . . . . . . . . . . . . . . . 19 ({(𝑃𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ↔ (((𝑃𝑥) = (𝑃𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))))
31 dff13 6995 . . . . . . . . . . . . . . . . . . . . 21 (𝑃:(0...(♯‘𝐹))–1-1𝑉 ↔ (𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏)))
32 elfzofz 13052 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹)))
33 elfzofz 13052 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ (0...(♯‘𝐹)))
34 fveqeq2 6658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = 𝑥 → ((𝑃𝑎) = (𝑃𝑏) ↔ (𝑃𝑥) = (𝑃𝑏)))
35 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = 𝑥 → (𝑎 = 𝑏𝑥 = 𝑏))
3634, 35imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = 𝑥 → (((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ↔ ((𝑃𝑥) = (𝑃𝑏) → 𝑥 = 𝑏)))
37 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑦 → (𝑃𝑏) = (𝑃𝑦))
3837eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑦 → ((𝑃𝑥) = (𝑃𝑏) ↔ (𝑃𝑥) = (𝑃𝑦)))
39 eqeq2 2813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑦 → (𝑥 = 𝑏𝑥 = 𝑦))
4038, 39imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 = 𝑦 → (((𝑃𝑥) = (𝑃𝑏) → 𝑥 = 𝑏) ↔ ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦)))
4136, 40rspc2v 3584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦)))
4232, 33, 41syl2an 598 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦)))
4342a1dd 50 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦))))
4443com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃𝑥) = (𝑃𝑦) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
4544adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃𝑥) = (𝑃𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
46 hashcl 13717 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
4732a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((♯‘𝐹) ∈ ℕ0 → (𝑥 ∈ (0..^(♯‘𝐹)) → 𝑥 ∈ (0...(♯‘𝐹))))
48 fzofzp1 13133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 ∈ (0..^(♯‘𝐹)) → (𝑦 + 1) ∈ (0...(♯‘𝐹)))
4947, 48anim12d1 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((♯‘𝐹) ∈ ℕ0 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹)))))
5049imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))))
51 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = (𝑦 + 1) → (𝑃𝑏) = (𝑃‘(𝑦 + 1)))
5251eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = (𝑦 + 1) → ((𝑃𝑥) = (𝑃𝑏) ↔ (𝑃𝑥) = (𝑃‘(𝑦 + 1))))
53 eqeq2 2813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = (𝑦 + 1) → (𝑥 = 𝑏𝑥 = (𝑦 + 1)))
5452, 53imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = (𝑦 + 1) → (((𝑃𝑥) = (𝑃𝑏) → 𝑥 = 𝑏) ↔ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
5536, 54rspc2v 3584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (0...(♯‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
5650, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
5756imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏)) → ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1)))
58 fzofzp1 13133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑥 ∈ (0..^(♯‘𝐹)) → (𝑥 + 1) ∈ (0...(♯‘𝐹)))
5958a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((♯‘𝐹) ∈ ℕ0 → (𝑥 ∈ (0..^(♯‘𝐹)) → (𝑥 + 1) ∈ (0...(♯‘𝐹))))
6059, 33anim12d1 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((♯‘𝐹) ∈ ℕ0 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹)))))
6160imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))))
62 fveqeq2 6658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = (𝑥 + 1) → ((𝑃𝑎) = (𝑃𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃𝑏)))
63 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = (𝑥 + 1) → (𝑎 = 𝑏 ↔ (𝑥 + 1) = 𝑏))
6462, 63imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑎 = (𝑥 + 1) → (((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃𝑏) → (𝑥 + 1) = 𝑏)))
6537eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑦 → ((𝑃‘(𝑥 + 1)) = (𝑃𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃𝑦)))
66 eqeq2 2813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑦 → ((𝑥 + 1) = 𝑏 ↔ (𝑥 + 1) = 𝑦))
6765, 66imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑦 → (((𝑃‘(𝑥 + 1)) = (𝑃𝑏) → (𝑥 + 1) = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃𝑦) → (𝑥 + 1) = 𝑦)))
6864, 67rspc2v 3584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑥 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝑦 ∈ (0...(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃‘(𝑥 + 1)) = (𝑃𝑦) → (𝑥 + 1) = 𝑦)))
6961, 68syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃‘(𝑥 + 1)) = (𝑃𝑦) → (𝑥 + 1) = 𝑦)))
7069imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏)) → ((𝑃‘(𝑥 + 1)) = (𝑃𝑦) → (𝑥 + 1) = 𝑦))
7157, 70anim12d 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏)) → (((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦)) → (𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦)))
7271expimpd 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ∧ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → (𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦)))
73 oveq1 7146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
7473eqeq1d 2803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
7574adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
76 elfzonn0 13081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℕ0)
77 nn0cn 11899 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0𝑦 ∈ ℂ)
78 add1p1 11880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℂ → ((𝑦 + 1) + 1) = (𝑦 + 2))
7977, 78syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 ∈ ℕ0 → ((𝑦 + 1) + 1) = (𝑦 + 2))
8079eqeq1d 2803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 ∈ ℕ0 → (((𝑦 + 1) + 1) = 𝑦 ↔ (𝑦 + 2) = 𝑦))
81 2cnd 11707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
82 2ne0 11733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 ≠ 0
8382a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → 2 ≠ 0)
84 addn0nid 11053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑦 + 2) ≠ 𝑦)
8577, 81, 83, 84syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 ∈ ℕ0 → (𝑦 + 2) ≠ 𝑦)
86 eqneqall 3001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 + 2) = 𝑦 → ((𝑦 + 2) ≠ 𝑦𝑥 = 𝑦))
8785, 86syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 ∈ ℕ0 → ((𝑦 + 2) = 𝑦𝑥 = 𝑦))
8880, 87sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 ∈ ℕ0 → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
8976, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 ∈ (0..^(♯‘𝐹)) → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9089adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9190adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9275, 91sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦𝑥 = 𝑦))
9392expimpd 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
9493adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
9572, 94syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((♯‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹)))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ∧ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → 𝑥 = 𝑦))
9695ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝐹) ∈ ℕ0 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ∧ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → 𝑥 = 𝑦)))
9746, 96syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ∧ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → 𝑥 = 𝑦)))
9897com3l 89 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ∧ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → (𝐹 ∈ Fin → 𝑥 = 𝑦)))
9998expd 419 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦)) → (𝐹 ∈ Fin → 𝑥 = 𝑦))))
10099com34 91 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → (((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦)) → 𝑥 = 𝑦))))
101100com14 96 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦)) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
10245, 101jaoi 854 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑃𝑥) = (𝑃𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → (∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
103102adantld 494 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑃𝑥) = (𝑃𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → ((𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑎 ∈ (0...(♯‘𝐹))∀𝑏 ∈ (0...(♯‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏)) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
10431, 103syl5bi 245 . . . . . . . . . . . . . . . . . . . 20 ((((𝑃𝑥) = (𝑃𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
105104com23 86 . . . . . . . . . . . . . . . . . . 19 ((((𝑃𝑥) = (𝑃𝑦) ∧ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1))) ∨ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) ∧ (𝑃‘(𝑥 + 1)) = (𝑃𝑦))) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
10630, 105sylbi 220 . . . . . . . . . . . . . . . . . 18 ({(𝑃𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
10725, 106syl 17 . . . . . . . . . . . . . . . . 17 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦))))
108107ex 416 . . . . . . . . . . . . . . . 16 ((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) → (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦)))))
10918, 108syl 17 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = (𝐹𝑦) → (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → 𝑥 = 𝑦)))))
110109com15 101 . . . . . . . . . . . . . 14 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
11117, 110syld 47 . . . . . . . . . . . . 13 ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐹 ∈ Fin → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
112111com14 96 . . . . . . . . . . . 12 (𝑃:(0...(♯‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
113112imp 410 . . . . . . . . . . 11 ((𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
114113impcom 411 . . . . . . . . . 10 ((𝐹 ∈ Fin ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → ((𝑥 ∈ (0..^(♯‘𝐹)) ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
115114ralrimivv 3158 . . . . . . . . 9 ((𝐹 ∈ Fin ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
116115adantlr 714 . . . . . . . 8 (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
117 dff13 6995 . . . . . . . 8 (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1184, 116, 117sylanbrc 586 . . . . . . 7 (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
119 df-f1 6333 . . . . . . 7 (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun 𝐹))
120118, 119sylib 221 . . . . . 6 (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun 𝐹))
121 simpr 488 . . . . . 6 ((𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun 𝐹) → Fun 𝐹)
122120, 121syl 17 . . . . 5 (((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) ∧ (𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → Fun 𝐹)
123122ex 416 . . . 4 ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → ((𝑃:(0...(♯‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → Fun 𝐹))
124123expd 419 . . 3 ((𝐹 ∈ Fin ∧ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹)))
1251, 2, 124syl2anc 587 . 2 (𝐹 ∈ Word dom 𝐼 → (𝑃:(0...(♯‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹)))
126125impcom 411 1 ((𝑃:(0...(♯‘𝐹))–1-1𝑉𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  {cpr 4530  ◡ccnv 5522  dom cdm 5523  Fun wfun 6322  ⟶wf 6324  –1-1→wf1 6325  ‘cfv 6328  (class class class)co 7139  Fincfn 8496  ℂcc 10528  0cc0 10530  1c1 10531   + caddc 10533  2c2 11684  ℕ0cn0 11889  ...cfz 12889  ..^cfzo 13032  ♯chash 13690  Word cword 13861 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862 This theorem is referenced by:  upgrwlkdvde  27529
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