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Theorem injresinjlem 13823
Description: Lemma for injresinj 13824. (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 13808 . . . . . . 7 ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2 fvinim0ffz 13822 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 3045 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑌 → (𝐹‘0) = (𝐹𝑌))
54eqcoms 2743 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 0 → (𝐹‘0) = (𝐹𝑌))
65eleq1d 2824 . . . . . . . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 12932 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 13723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 13715 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13sstrdi 4008 . . . . . . . . . . . . . . . . . . . . . . 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 3070 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌))
19 fveqeq2 6916 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
37 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑌 → (𝐹𝐾) = (𝐹𝑌))
3837eqcoms 2743 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 𝐾 → (𝐹𝐾) = (𝐹𝑌))
3938eleq1d 2824 . . . . . . . . . . . . . . . . . . . . 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 857 . . . . . . . . . . . . . 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 13808 . . . . . . . . . . 11 ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53 eqtr3 2761 . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑋 → (𝐹𝐾) = (𝐹𝑋))
5958eqcoms 2743 . . . . . . . . . . . . . . . . 17 (𝑋 = 𝐾 → (𝐹𝐾) = (𝐹𝑋))
6059adantr 480 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹𝐾) = (𝐹𝑋))
6157, 60neeq12d 3000 . . . . . . . . . . . . . . 15 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
62 df-ne 2939 . . . . . . . . . . . . . . . 16 ((𝐹𝑌) ≠ (𝐹𝑋) ↔ ¬ (𝐹𝑌) = (𝐹𝑋))
63 pm2.24 124 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑌) = (𝐹𝑋) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6463eqcoms 2743 . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . 18 (0 = 𝑋 → (𝐹‘0) = (𝐹𝑋))
7069eqcoms 2743 . . . . . . . . . . . . . . . . 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 3000 . . . . . . . . . . . . . . 15 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
74 df-ne 2939 . . . . . . . . . . . . . . . 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 2761 . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . 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 847   = wceq 1537  wcel 2106  wne 2938  wnel 3044  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  {cpr 4633  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  0cn0 12524  cuz 12876  ...cfz 13544  ..^cfzo 13691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692
This theorem is referenced by:  injresinj  13824
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