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Theorem injresinjlem 13692
Description: Lemma for injresinj 13693. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-2021.) (Revised by Thierry Arnoux, 23-Dec-2021.)
Assertion
Ref Expression
injresinjlem 𝑌 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))

Proof of Theorem injresinjlem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elfznelfzo 13677 . . . . . . 7 ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2 fvinim0ffz 13691 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 3050 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑌 → (𝐹‘0) = (𝐹𝑌))
54eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 0 → (𝐹‘0) = (𝐹𝑌))
65eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
76notbid 317 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
87biimpd 228 . . . . . . . . . . . . . . . . . . 19 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
9 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 12817 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 13599 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 13591 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13sstrdi 3956 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6914 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
169, 14, 15syl2an 596 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
1716notbid 317 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
18 ralnex 3075 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌))
19 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑌) ↔ (𝐹𝑋) = (𝐹𝑌)))
2019notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑋 → (¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌)))
2120rspcva 3579 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑋 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌)) → ¬ (𝐹𝑋) = (𝐹𝑌))
22 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (¬ (𝐹𝑋) = (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
2322a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (¬ (𝐹𝑋) = (𝐹𝑌) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
24232a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (¬ (𝐹𝑋) = (𝐹𝑌) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
2521, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌)) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
2625expcom 414 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) → (𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
2726com24 95 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
2818, 27sylbir 234 . . . . . . . . . . . . . . . . . . . . . 22 (¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
2928com12 32 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
3017, 29sylbid 239 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
3130com12 32 . . . . . . . . . . . . . . . . . . 19 (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
328, 31syl6com 37 . . . . . . . . . . . . . . . . . 18 (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → (𝑌 = 0 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
333, 32sylbi 216 . . . . . . . . . . . . . . . . 17 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) → (𝑌 = 0 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
3433adantr 481 . . . . . . . . . . . . . . . 16 (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))) → (𝑌 = 0 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
3534com12 32 . . . . . . . . . . . . . . 15 (𝑌 = 0 → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
36 df-nel 3050 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
37 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑌 → (𝐹𝐾) = (𝐹𝑌))
3837eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 𝐾 → (𝐹𝐾) = (𝐹𝑌))
3938eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 𝐾 → ((𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4039notbid 317 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4140biimpd 228 . . . . . . . . . . . . . . . . . . 19 (𝑌 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4241, 31syl6com 37 . . . . . . . . . . . . . . . . . 18 (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) → (𝑌 = 𝐾 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
4336, 42sylbi 216 . . . . . . . . . . . . . . . . 17 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) → (𝑌 = 𝐾 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
4443adantl 482 . . . . . . . . . . . . . . . 16 (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))) → (𝑌 = 𝐾 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
4544com12 32 . . . . . . . . . . . . . . 15 (𝑌 = 𝐾 → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
4635, 45jaoi 855 . . . . . . . . . . . . . 14 ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
4746com13 88 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
482, 47sylbid 239 . . . . . . . . . . . 12 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
4948com14 96 . . . . . . . . . . 11 (𝑋 ∈ (0...𝐾) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
5049com12 32 . . . . . . . . . 10 (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
5150com15 101 . . . . . . . . 9 (𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
52 elfznelfzo 13677 . . . . . . . . . . 11 ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53 eqtr3 2762 . . . . . . . . . . . . . 14 ((𝑋 = 0 ∧ 𝑌 = 0) → 𝑋 = 𝑌)
54 2a1 28 . . . . . . . . . . . . . . 15 (𝑋 = 𝑌 → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
55542a1d 26 . . . . . . . . . . . . . 14 (𝑋 = 𝑌 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
5653, 55syl 17 . . . . . . . . . . . . 13 ((𝑋 = 0 ∧ 𝑌 = 0) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
575adantl 482 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹‘0) = (𝐹𝑌))
58 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑋 → (𝐹𝐾) = (𝐹𝑋))
5958eqcoms 2744 . . . . . . . . . . . . . . . . 17 (𝑋 = 𝐾 → (𝐹𝐾) = (𝐹𝑋))
6059adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹𝐾) = (𝐹𝑋))
6157, 60neeq12d 3005 . . . . . . . . . . . . . . 15 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
62 df-ne 2944 . . . . . . . . . . . . . . . 16 ((𝐹𝑌) ≠ (𝐹𝑋) ↔ ¬ (𝐹𝑌) = (𝐹𝑋))
63 pm2.24 124 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑌) = (𝐹𝑋) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6463eqcoms 2744 . . . . . . . . . . . . . . . . 17 ((𝐹𝑋) = (𝐹𝑌) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6564com12 32 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑌) = (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6662, 65sylbi 216 . . . . . . . . . . . . . . 15 ((𝐹𝑌) ≠ (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6761, 66syl6bi 252 . . . . . . . . . . . . . 14 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
68672a1d 26 . . . . . . . . . . . . 13 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
69 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (0 = 𝑋 → (𝐹‘0) = (𝐹𝑋))
7069eqcoms 2744 . . . . . . . . . . . . . . . . 17 (𝑋 = 0 → (𝐹‘0) = (𝐹𝑋))
7170adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘0) = (𝐹𝑋))
7238adantl 482 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹𝐾) = (𝐹𝑌))
7371, 72neeq12d 3005 . . . . . . . . . . . . . . 15 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
74 df-ne 2944 . . . . . . . . . . . . . . . 16 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌))
7574, 22sylbi 216 . . . . . . . . . . . . . . 15 ((𝐹𝑋) ≠ (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
7673, 75syl6bi 252 . . . . . . . . . . . . . 14 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
77762a1d 26 . . . . . . . . . . . . 13 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
78 eqtr3 2762 . . . . . . . . . . . . . 14 ((𝑋 = 𝐾𝑌 = 𝐾) → 𝑋 = 𝑌)
7978, 55syl 17 . . . . . . . . . . . . 13 ((𝑋 = 𝐾𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
8056, 68, 77, 79ccase 1036 . . . . . . . . . . . 12 (((𝑋 = 0 ∨ 𝑋 = 𝐾) ∧ (𝑌 = 0 ∨ 𝑌 = 𝐾)) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
8180ex 413 . . . . . . . . . . 11 ((𝑋 = 0 ∨ 𝑋 = 𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
8252, 81syl 17 . . . . . . . . . 10 ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
8382expcom 414 . . . . . . . . 9 𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
8451, 83pm2.61i 182 . . . . . . . 8 (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
8584com12 32 . . . . . . 7 ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
861, 85syl 17 . . . . . 6 ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
8786ex 413 . . . . 5 (𝑌 ∈ (0...𝐾) → (¬ 𝑌 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
8887com23 86 . . . 4 (𝑌 ∈ (0...𝐾) → (𝑋 ∈ (0...𝐾) → (¬ 𝑌 ∈ (1..^𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))))
8988impcom 408 . . 3 ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → (¬ 𝑌 ∈ (1..^𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
9089com12 32 . 2 𝑌 ∈ (1..^𝐾) → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
9190com25 99 1 𝑌 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wnel 3049  wral 3064  wrex 3073  cin 3909  wss 3910  c0 4282  {cpr 4588  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052  0cn0 12413  cuz 12763  ...cfz 13424  ..^cfzo 13567
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568
This theorem is referenced by:  injresinj  13693
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