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Theorem upgrwlkdvdelem 26688
Description: Lemma for upgrwlkdvde 26689. (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 13355 . . 3 (𝐹 ∈ Word dom 𝐼𝐹 ∈ Fin)
2 wrdf 13342 . . 3 (𝐹 ∈ Word dom 𝐼𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
3 simpr 476 . . . . . . . . 9 ((𝐹 ∈ Fin ∧ 𝐹:(0..^(#‘𝐹))⟶dom 𝐼) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
43adantr 480 . . . . . . . 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 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
65fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → (𝐼‘(𝐹𝑘)) = (𝐼‘(𝐹𝑥)))
7 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑃𝑘) = (𝑃𝑥))
8 oveq1 6697 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑥 → (𝑘 + 1) = (𝑥 + 1))
98fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
107, 9preq12d 4308 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
116, 10eqeq12d 2666 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑥 → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
1211rspcv 3336 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
13 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
1413fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → (𝐼‘(𝐹𝑘)) = (𝐼‘(𝐹𝑦)))
15 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑃𝑘) = (𝑃𝑦))
16 oveq1 6697 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑦 → (𝑘 + 1) = (𝑦 + 1))
1716fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
1815, 17preq12d 4308 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
1914, 18eqeq12d 2666 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
2019rspcv 3336 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
2112, 20anim12ii 593 . . . . . . . . . . . . . 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))})))
22 fveq2 6229 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = (𝐹𝑦) → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)))
23 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
2423eqcomd 2657 . . . . . . . . . . . . . . . . . . . 20 (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = (𝐼‘(𝐹𝑥)))
2524adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = (𝐼‘(𝐹𝑥)))
26 simpl 472 . . . . . . . . . . . . . . . . . . 19 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)))
27 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
2827adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑦)) ∧ ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ∧ (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})) → (𝐼‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
2925, 26, 283eqtrd 2689 . . . . . . . . . . . . . . . . . 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))})
30 fvex 6239 . . . . . . . . . . . . . . . . . . . 20 (𝑃𝑥) ∈ V
31 fvex 6239 . . . . . . . . . . . . . . . . . . . 20 (𝑃‘(𝑥 + 1)) ∈ V
32 fvex 6239 . . . . . . . . . . . . . . . . . . . 20 (𝑃𝑦) ∈ V
33 fvex 6239 . . . . . . . . . . . . . . . . . . . 20 (𝑃‘(𝑦 + 1)) ∈ V
3430, 31, 32, 33preq12b 4413 . . . . . . . . . . . . . . . . . . 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))))
35 dff13 6552 . . . . . . . . . . . . . . . . . . . . 21 (𝑃:(0...(#‘𝐹))–1-1𝑉 ↔ (𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑎 ∈ (0...(#‘𝐹))∀𝑏 ∈ (0...(#‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏)))
36 elfzofz 12524 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^(#‘𝐹)) → 𝑥 ∈ (0...(#‘𝐹)))
37 elfzofz 12524 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ (0...(#‘𝐹)))
38 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 = 𝑥 → (𝑃𝑎) = (𝑃𝑥))
3938eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = 𝑥 → ((𝑃𝑎) = (𝑃𝑏) ↔ (𝑃𝑥) = (𝑃𝑏)))
40 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = 𝑥 → (𝑎 = 𝑏𝑥 = 𝑏))
4139, 40imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = 𝑥 → (((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ↔ ((𝑃𝑥) = (𝑃𝑏) → 𝑥 = 𝑏)))
42 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑦 → (𝑃𝑏) = (𝑃𝑦))
4342eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑦 → ((𝑃𝑥) = (𝑃𝑏) ↔ (𝑃𝑥) = (𝑃𝑦)))
44 eqeq2 2662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑦 → (𝑥 = 𝑏𝑥 = 𝑦))
4543, 44imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 = 𝑦 → (((𝑃𝑥) = (𝑃𝑏) → 𝑥 = 𝑏) ↔ ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦)))
4641, 45rspc2v 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹))) → (∀𝑎 ∈ (0...(#‘𝐹))∀𝑏 ∈ (0...(#‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦)))
4736, 37, 46syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (∀𝑎 ∈ (0...(#‘𝐹))∀𝑏 ∈ (0...(#‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦)))
4847a1dd 50 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (∀𝑎 ∈ (0...(#‘𝐹))∀𝑏 ∈ (0...(#‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑃𝑥) = (𝑃𝑦) → 𝑥 = 𝑦))))
4948com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃𝑥) = (𝑃𝑦) → (∀𝑎 ∈ (0...(#‘𝐹))∀𝑏 ∈ (0...(#‘𝐹))((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) → (𝐹 ∈ Fin → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → 𝑥 = 𝑦))))
5049adantr 480 . . . . . . . . . . . . . . . . . . . . . . 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))))
51 hashcl 13185 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹 ∈ Fin → (#‘𝐹) ∈ ℕ0)
5236a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝐹) ∈ ℕ0 → (𝑥 ∈ (0..^(#‘𝐹)) → 𝑥 ∈ (0...(#‘𝐹))))
53 fzofzp1 12605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 ∈ (0..^(#‘𝐹)) → (𝑦 + 1) ∈ (0...(#‘𝐹)))
5452, 53anim12d1 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝐹) ∈ ℕ0 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝑥 ∈ (0...(#‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(#‘𝐹)))))
5554imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → (𝑥 ∈ (0...(#‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(#‘𝐹))))
56 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = (𝑦 + 1) → (𝑃𝑏) = (𝑃‘(𝑦 + 1)))
5756eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = (𝑦 + 1) → ((𝑃𝑥) = (𝑃𝑏) ↔ (𝑃𝑥) = (𝑃‘(𝑦 + 1))))
58 eqeq2 2662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = (𝑦 + 1) → (𝑥 = 𝑏𝑥 = (𝑦 + 1)))
5957, 58imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = (𝑦 + 1) → (((𝑃𝑥) = (𝑃𝑏) → 𝑥 = 𝑏) ↔ ((𝑃𝑥) = (𝑃‘(𝑦 + 1)) → 𝑥 = (𝑦 + 1))))
6041, 59rspc2v 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
6155, 60syl 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))))
6261imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
63 fzofzp1 12605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑥 ∈ (0..^(#‘𝐹)) → (𝑥 + 1) ∈ (0...(#‘𝐹)))
6463a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝐹) ∈ ℕ0 → (𝑥 ∈ (0..^(#‘𝐹)) → (𝑥 + 1) ∈ (0...(#‘𝐹))))
6564, 37anim12d1 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝐹) ∈ ℕ0 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝑥 + 1) ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹)))))
6665imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → ((𝑥 + 1) ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹))))
67 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎 = (𝑥 + 1) → (𝑃𝑎) = (𝑃‘(𝑥 + 1)))
6867eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = (𝑥 + 1) → ((𝑃𝑎) = (𝑃𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃𝑏)))
69 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = (𝑥 + 1) → (𝑎 = 𝑏 ↔ (𝑥 + 1) = 𝑏))
7068, 69imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑎 = (𝑥 + 1) → (((𝑃𝑎) = (𝑃𝑏) → 𝑎 = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃𝑏) → (𝑥 + 1) = 𝑏)))
7142eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑦 → ((𝑃‘(𝑥 + 1)) = (𝑃𝑏) ↔ (𝑃‘(𝑥 + 1)) = (𝑃𝑦)))
72 eqeq2 2662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑦 → ((𝑥 + 1) = 𝑏 ↔ (𝑥 + 1) = 𝑦))
7371, 72imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑦 → (((𝑃‘(𝑥 + 1)) = (𝑃𝑏) → (𝑥 + 1) = 𝑏) ↔ ((𝑃‘(𝑥 + 1)) = (𝑃𝑦) → (𝑥 + 1) = 𝑦)))
7470, 73rspc2v 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
7566, 74syl 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)))
7675imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
7762, 76anim12d 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
7877expimpd 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
79 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8079eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
8180adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦 ↔ ((𝑦 + 1) + 1) = 𝑦))
82 elfzonn0 12552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ ℕ0)
83 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0𝑦 ∈ ℂ)
84 add1p1 11321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℂ → ((𝑦 + 1) + 1) = (𝑦 + 2))
8583, 84syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 ∈ ℕ0 → ((𝑦 + 1) + 1) = (𝑦 + 2))
8685eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 ∈ ℕ0 → (((𝑦 + 1) + 1) = 𝑦 ↔ (𝑦 + 2) = 𝑦))
87 2cnd 11131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
88 2ne0 11151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 ≠ 0
8988a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → 2 ≠ 0)
90 addn0nid 10489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑦 + 2) ≠ 𝑦)
9183, 87, 89, 90syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 ∈ ℕ0 → (𝑦 + 2) ≠ 𝑦)
92 eqneqall 2834 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 + 2) = 𝑦 → ((𝑦 + 2) ≠ 𝑦𝑥 = 𝑦))
9391, 92syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 ∈ ℕ0 → ((𝑦 + 2) = 𝑦𝑥 = 𝑦))
9486, 93sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 ∈ ℕ0 → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9582, 94syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 ∈ (0..^(#‘𝐹)) → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9695adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9796adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → (((𝑦 + 1) + 1) = 𝑦𝑥 = 𝑦))
9881, 97sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑥 = (𝑦 + 1)) → ((𝑥 + 1) = 𝑦𝑥 = 𝑦))
9998expimpd 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
10099adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((#‘𝐹) ∈ ℕ0 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → ((𝑥 = (𝑦 + 1) ∧ (𝑥 + 1) = 𝑦) → 𝑥 = 𝑦))
10178, 100syld 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))
102101ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
10351, 102syl 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)))
104103com3l 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)))
105104expd 451 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
106105com34 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))))
107106com14 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))))
10850, 107jaoi 393 . . . . . . . . . . . . . . . . . . . . . 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))))
109108adantld 482 . . . . . . . . . . . . . . . . . . . . 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))))
11035, 109syl5bi 232 . . . . . . . . . . . . . . . . . . . 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))))
111110com23 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))))
11234, 111sylbi 207 . . . . . . . . . . . . . . . . . 18 ({(𝑃𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → (𝐹 ∈ Fin → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → 𝑥 = 𝑦))))
11329, 112syl 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))))
114113ex 449 . . . . . . . . . . . . . . . 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)))))
11522, 114syl 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)))))
116115com15 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)))))
11721, 116syld 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)))))
118117com14 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)))))
119118imp 444 . . . . . . . . . . 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))))
120119impcom 445 . . . . . . . . . 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)))
121120ralrimivv 2999 . . . . . . . . 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))
122121adantlr 751 . . . . . . . 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))
123 dff13 6552 . . . . . . . 8 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1244, 122, 123sylanbrc 699 . . . . . . 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)
125 df-f1 5931 . . . . . . 7 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 ∧ Fun 𝐹))
126124, 125sylib 208 . . . . . 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))
127 simpr 476 . . . . . 6 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐼 ∧ Fun 𝐹) → Fun 𝐹)
128126, 127syl 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)
129128ex 449 . . . 4 ((𝐹 ∈ Fin ∧ 𝐹:(0..^(#‘𝐹))⟶dom 𝐼) → ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → Fun 𝐹))
130129expd 451 . . 3 ((𝐹 ∈ Fin ∧ 𝐹:(0..^(#‘𝐹))⟶dom 𝐼) → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹)))
1311, 2, 130syl2anc 694 . 2 (𝐹 ∈ Word dom 𝐼 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹)))
132131impcom 445 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 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {cpr 4212  ccnv 5142  dom cdm 5143  Fun wfun 5920  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  0cc0 9974  1c1 9975   + caddc 9977  2c2 11108  0cn0 11330  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331
This theorem is referenced by:  upgrwlkdvde  26689
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