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Theorem injresinjlem 13826
Description: Lemma for injresinj 13827. (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 13811 . . . . . . 7 ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2 fvinim0ffz 13825 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 3047 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑌 → (𝐹‘0) = (𝐹𝑌))
54eqcoms 2745 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 0 → (𝐹‘0) = (𝐹𝑌))
65eleq1d 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
76notbid 318 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
87biimpd 229 . . . . . . . . . . . . . . . . . . 19 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
9 ffn 6736 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 12934 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 13726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 13718 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13sstrdi 3996 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6981 . . . . . . . . . . . . . . . . . . . . . . 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 318 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
18 ralnex 3072 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌))
19 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑌) ↔ (𝐹𝑋) = (𝐹𝑌)))
2019notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑋 → (¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌)))
2120rspcva 3620 . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 3047 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
37 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑌 → (𝐹𝐾) = (𝐹𝑌))
3837eqcoms 2745 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 𝐾 → (𝐹𝐾) = (𝐹𝑌))
3938eleq1d 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 𝐾 → ((𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4039notbid 318 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4140biimpd 229 . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . 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 240 . . . . . . . . . . . 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 13811 . . . . . . . . . . 11 ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53 eqtr3 2763 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹‘0) = (𝐹𝑌))
58 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑋 → (𝐹𝐾) = (𝐹𝑋))
5958eqcoms 2745 . . . . . . . . . . . . . . . . 17 (𝑋 = 𝐾 → (𝐹𝐾) = (𝐹𝑋))
6059adantr 480 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹𝐾) = (𝐹𝑋))
6157, 60neeq12d 3002 . . . . . . . . . . . . . . 15 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
62 df-ne 2941 . . . . . . . . . . . . . . . 16 ((𝐹𝑌) ≠ (𝐹𝑋) ↔ ¬ (𝐹𝑌) = (𝐹𝑋))
63 pm2.24 124 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑌) = (𝐹𝑋) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6463eqcoms 2745 . . . . . . . . . . . . . . . . 17 ((𝐹𝑋) = (𝐹𝑌) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6564com12 32 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑌) = (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6662, 65sylbi 217 . . . . . . . . . . . . . . 15 ((𝐹𝑌) ≠ (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6761, 66biimtrdi 253 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . 18 (0 = 𝑋 → (𝐹‘0) = (𝐹𝑋))
7069eqcoms 2745 . . . . . . . . . . . . . . . . 17 (𝑋 = 0 → (𝐹‘0) = (𝐹𝑋))
7170adantr 480 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘0) = (𝐹𝑋))
7238adantl 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹𝐾) = (𝐹𝑌))
7371, 72neeq12d 3002 . . . . . . . . . . . . . . 15 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
74 df-ne 2941 . . . . . . . . . . . . . . . 16 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌))
7574, 22sylbi 217 . . . . . . . . . . . . . . 15 ((𝐹𝑋) ≠ (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
7673, 75biimtrdi 253 . . . . . . . . . . . . . 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 2763 . . . . . . . . . . . . . 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 1038 . . . . . . . . . . . 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 412 . . . . . . . . . . 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 413 . . . . . . . . 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 412 . . . . 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 407 . . 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 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333  {cpr 4628  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  0cn0 12526  cuz 12878  ...cfz 13547  ..^cfzo 13694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695
This theorem is referenced by:  injresinj  13827
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