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Theorem injresinjlem 12780
 Description: Lemma for injresinj 12781. (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 12765 . . . . . . 7 ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2 fvinim0ffz 12779 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 3034 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6350 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑌 → (𝐹‘0) = (𝐹𝑌))
54eqcoms 2766 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 0 → (𝐹‘0) = (𝐹𝑌))
65eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
76notbid 307 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
87biimpd 219 . . . . . . . . . . . . . . . . . . 19 (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
9 ffn 6204 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 11923 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 12687 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 12680 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13syl6ss 3754 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6413 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
169, 14, 15syl2an 495 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
1716notbid 307 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌)))
18 ralnex 3128 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌))
19 fveq2 6350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑋 → (𝐹𝑧) = (𝐹𝑋))
2019eqeq1d 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑌) ↔ (𝐹𝑋) = (𝐹𝑌)))
2120notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑋 → (¬ (𝐹𝑧) = (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌)))
2221rspcva 3445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑋 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌)) → ¬ (𝐹𝑋) = (𝐹𝑌))
23 pm2.21 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (¬ (𝐹𝑋) = (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
2423a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (¬ (𝐹𝑋) = (𝐹𝑌) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
25242a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (¬ (𝐹𝑋) = (𝐹𝑌) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
2622, 25syl 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)))))
2726expcom 450 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑌) → (𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
2827com24 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))))))
2918, 28sylbir 225 . . . . . . . . . . . . . . . . . . . . . 22 (¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑌) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
3029com12 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))))))
3117, 30sylbid 230 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
3231com12 32 . . . . . . . . . . . . . . . . . . 19 (¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾)) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
338, 32syl6com 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)))))))
343, 33sylbi 207 . . . . . . . . . . . . . . . . 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)))))))
3534adantr 472 . . . . . . . . . . . . . . . 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)))))))
3635com12 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)))))))
37 df-nel 3034 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
38 fveq2 6350 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑌 → (𝐹𝐾) = (𝐹𝑌))
3938eqcoms 2766 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 = 𝐾 → (𝐹𝐾) = (𝐹𝑌))
4039eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 = 𝐾 → ((𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4140notbid 307 . . . . . . . . . . . . . . . . . . . 20 (𝑌 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4241biimpd 219 . . . . . . . . . . . . . . . . . . 19 (𝑌 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑌) ∈ (𝐹 “ (1..^𝐾))))
4342, 32syl6com 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)))))))
4437, 43sylbi 207 . . . . . . . . . . . . . . . . 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)))))))
4544adantl 473 . . . . . . . . . . . . . . . 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)))))))
4645com12 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)))))))
4736, 46jaoi 393 . . . . . . . . . . . . . 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)))))))
4847com13 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)))))))
492, 48sylbid 230 . . . . . . . . . . . 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)))))))
5049com14 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)))))))
5150com12 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)))))))
5251com15 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)))))))
53 elfznelfzo 12765 . . . . . . . . . . 11 ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
54 eqtr3 2779 . . . . . . . . . . . . . 14 ((𝑋 = 0 ∧ 𝑌 = 0) → 𝑋 = 𝑌)
55 2a1 28 . . . . . . . . . . . . . . 15 (𝑋 = 𝑌 → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
56552a1d 26 . . . . . . . . . . . . . 14 (𝑋 = 𝑌 → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
5754, 56syl 17 . . . . . . . . . . . . 13 ((𝑋 = 0 ∧ 𝑌 = 0) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
585adantl 473 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹‘0) = (𝐹𝑌))
59 fveq2 6350 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑋 → (𝐹𝐾) = (𝐹𝑋))
6059eqcoms 2766 . . . . . . . . . . . . . . . . 17 (𝑋 = 𝐾 → (𝐹𝐾) = (𝐹𝑋))
6160adantr 472 . . . . . . . . . . . . . . . 16 ((𝑋 = 𝐾𝑌 = 0) → (𝐹𝐾) = (𝐹𝑋))
6258, 61neeq12d 2991 . . . . . . . . . . . . . . 15 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
63 df-ne 2931 . . . . . . . . . . . . . . . 16 ((𝐹𝑌) ≠ (𝐹𝑋) ↔ ¬ (𝐹𝑌) = (𝐹𝑋))
64 pm2.24 121 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑌) = (𝐹𝑋) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6564eqcoms 2766 . . . . . . . . . . . . . . . . 17 ((𝐹𝑋) = (𝐹𝑌) → (¬ (𝐹𝑌) = (𝐹𝑋) → 𝑋 = 𝑌))
6665com12 32 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑌) = (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6763, 66sylbi 207 . . . . . . . . . . . . . . 15 ((𝐹𝑌) ≠ (𝐹𝑋) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
6862, 67syl6bi 243 . . . . . . . . . . . . . 14 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
69682a1d 26 . . . . . . . . . . . . 13 ((𝑋 = 𝐾𝑌 = 0) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
70 fveq2 6350 . . . . . . . . . . . . . . . . . 18 (0 = 𝑋 → (𝐹‘0) = (𝐹𝑋))
7170eqcoms 2766 . . . . . . . . . . . . . . . . 17 (𝑋 = 0 → (𝐹‘0) = (𝐹𝑋))
7271adantr 472 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘0) = (𝐹𝑋))
7339adantl 473 . . . . . . . . . . . . . . . 16 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹𝐾) = (𝐹𝑌))
7472, 73neeq12d 2991 . . . . . . . . . . . . . . 15 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
75 df-ne 2931 . . . . . . . . . . . . . . . 16 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ¬ (𝐹𝑋) = (𝐹𝑌))
7675, 23sylbi 207 . . . . . . . . . . . . . . 15 ((𝐹𝑋) ≠ (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
7774, 76syl6bi 243 . . . . . . . . . . . . . 14 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
78772a1d 26 . . . . . . . . . . . . 13 ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
79 eqtr3 2779 . . . . . . . . . . . . . 14 ((𝑋 = 𝐾𝑌 = 𝐾) → 𝑋 = 𝑌)
8079, 56syl 17 . . . . . . . . . . . . 13 ((𝑋 = 𝐾𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))))
8157, 69, 78, 80ccase 1024 . . . . . . . . . . . 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)))))
8281ex 449 . . . . . . . . . . 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))))))
8353, 82syl 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))))))
8483expcom 450 . . . . . . . . 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)))))))
8552, 84pm2.61i 176 . . . . . . . 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))))))
8685com12 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))))))
871, 86syl 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))))))
8887ex 449 . . . . 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)))))))
8988com23 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)))))))
9089impcom 445 . . 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))))))
9190com12 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))))))
9291com25 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 196   ∨ wo 382   ∧ wa 383   = wceq 1630   ∈ wcel 2137   ≠ wne 2930   ∉ wnel 3033  ∀wral 3048  ∃wrex 3049   ∩ cin 3712   ⊆ wss 3713  ∅c0 4056  {cpr 4321   “ cima 5267   Fn wfn 6042  ⟶wf 6043  ‘cfv 6047  (class class class)co 6811  0cc0 10126  1c1 10127  ℕ0cn0 11482  ℤ≥cuz 11877  ...cfz 12517  ..^cfzo 12657 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-fzo 12658 This theorem is referenced by:  injresinj  12781
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