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Theorem injresinjlem 13011
Description: Lemma for injresinj 13012. (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 12996 . . . . . . 7 ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2 fvinim0ffz 13010 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 3093 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6545 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑌 → (𝐹‘0) = (𝐹𝑌))
54eqcoms 2805 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 0 → (𝐹‘0) = (𝐹𝑌))
65eleq1d 2869 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
76notbid 319 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
87biimpd 230 . . . . . . . . . . . . . . . . . . 19 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
9 ffn 6389 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 12145 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 12918 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 12910 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13syl6ss 3907 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6612 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
169, 14, 15syl2an 595 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
1716notbid 319 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
18 ralnex 3202 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌))
19 fveqeq2 6554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑌) ↔ (𝐹𝑋) = (𝐹𝑌)))
2019notbid 319 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑋 → (¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌)))
2120rspcva 3559 . . . . . . . . . . . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 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 3093 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
37 fveq2 6545 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑌 → (𝐹𝐾) = (𝐹𝑌))
3837eqcoms 2805 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 𝐾 → (𝐹𝐾) = (𝐹𝑌))
3938eleq1d 2869 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 𝐾 → ((𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4039notbid 319 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4140biimpd 230 . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 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 852 . . . . . . . . . . . . . 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 241 . . . . . . . . . . . 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 12996 . . . . . . . . . . 11 ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53 eqtr3 2820 . . . . . . . . . . . . . 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 6545 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑋 → (𝐹𝐾) = (𝐹𝑋))
5958eqcoms 2805 . . . . . . . . . . . . . . . . 17 (𝑋 = 𝐾 → (𝐹𝐾) = (𝐹𝑋))
6059adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹𝐾) = (𝐹𝑋))
6157, 60neeq12d 3047 . . . . . . . . . . . . . . 15 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
62 df-ne 2987 . . . . . . . . . . . . . . . 16 ((𝐹𝑌) ≠ (𝐹𝑋) ↔ ¬ (𝐹𝑌) = (𝐹𝑋))
63 pm2.24 124 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑌) = (𝐹𝑋) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6463eqcoms 2805 . . . . . . . . . . . . . . . . 17 ((𝐹𝑋) = (𝐹𝑌) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6564com12 32 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑌) = (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6662, 65sylbi 218 . . . . . . . . . . . . . . 15 ((𝐹𝑌) ≠ (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6761, 66syl6bi 254 . . . . . . . . . . . . . 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 6545 . . . . . . . . . . . . . . . . . 18 (0 = 𝑋 → (𝐹‘0) = (𝐹𝑋))
7069eqcoms 2805 . . . . . . . . . . . . . . . . 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 3047 . . . . . . . . . . . . . . 15 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
74 df-ne 2987 . . . . . . . . . . . . . . . 16 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌))
7574, 22sylbi 218 . . . . . . . . . . . . . . 15 ((𝐹𝑋) ≠ (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
7673, 75syl6bi 254 . . . . . . . . . . . . . 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 2820 . . . . . . . . . . . . . 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 1030 . . . . . . . . . . . 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 183 . . . . . . . 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 207  wa 396  wo 842   = wceq 1525  wcel 2083  wne 2986  wnel 3092  wral 3107  wrex 3108  cin 3864  wss 3865  c0 4217  {cpr 4480  cima 5453   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  0cc0 10390  1c1 10391  0cn0 11751  cuz 12097  ...cfz 12746  ..^cfzo 12887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098  df-fz 12747  df-fzo 12888
This theorem is referenced by:  injresinj  13012
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