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Theorem injresinjlem 12405
Description: Lemma for injresinj 12406. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-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 12394 . . . . . . 7 ((𝑦 ∈ (0...𝐾) ∧ ¬ 𝑦 ∈ (1..^𝐾)) → (𝑦 = 0 ∨ 𝑦 = 𝐾))
2 fvinim0ffz 12404 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 2782 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑦 → (𝐹‘0) = (𝐹𝑦))
54eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 0 → (𝐹‘0) = (𝐹𝑦))
65eleq1d 2671 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
76notbid 306 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
87biimpd 217 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
9 ffn 5944 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 11564 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 12319 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 12312 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13syl6ss 3579 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6148 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
169, 14, 15syl2an 492 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
1716notbid 306 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
18 ralnex 2974 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑦) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦))
19 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
2120notbid 306 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦)))
2221rspcva 3279 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑦)) → ¬ (𝐹𝑥) = (𝐹𝑦))
23 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . 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 92 . . . . . . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . . . . . . 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 36 . . . . . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
38 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑦 → (𝐹𝐾) = (𝐹𝑦))
3938eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝐾 → (𝐹𝐾) = (𝐹𝑦))
4039eleq1d 2671 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐾 → ((𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
4140notbid 306 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
4241biimpd 217 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐾 → (¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
4342, 32syl6com 36 . . . . . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 392 . . . . . . . . . . . . . 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 85 . . . . . . . . . . . . 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 228 . . . . . . . . . . . 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 93 . . . . . . . . . . 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 98 . . . . . . . . 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 12394 . . . . . . . . . . 11 ((𝑥 ∈ (0...𝐾) ∧ ¬ 𝑥 ∈ (1..^𝐾)) → (𝑥 = 0 ∨ 𝑥 = 𝐾))
54 eqtr3 2630 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐾𝑦 = 0) → (𝐹‘0) = (𝐹𝑦))
59 fveq2 6088 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑥 → (𝐹𝐾) = (𝐹𝑥))
6059eqcoms 2617 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐾 → (𝐹𝐾) = (𝐹𝑥))
6160adantr 479 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐾𝑦 = 0) → (𝐹𝐾) = (𝐹𝑥))
6258, 61neeq12d 2842 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐾𝑦 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑦) ≠ (𝐹𝑥)))
63 df-ne 2781 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ≠ (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
64 pm2.24 119 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) = (𝐹𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) → 𝑥 = 𝑦))
6564eqcoms 2617 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = (𝐹𝑦) → (¬ (𝐹𝑦) = (𝐹𝑥) → 𝑥 = 𝑦))
6665com12 32 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑦) = (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6763, 66sylbi 205 . . . . . . . . . . . . . . 15 ((𝐹𝑦) ≠ (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6862, 67syl6bi 241 . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . . 18 (0 = 𝑥 → (𝐹‘0) = (𝐹𝑥))
7170eqcoms 2617 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝐹‘0) = (𝐹𝑥))
7271adantr 479 . . . . . . . . . . . . . . . 16 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘0) = (𝐹𝑥))
7339adantl 480 . . . . . . . . . . . . . . . 16 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹𝐾) = (𝐹𝑦))
7472, 73neeq12d 2842 . . . . . . . . . . . . . . 15 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑥) ≠ (𝐹𝑦)))
75 df-ne 2781 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) ≠ (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
7675, 23sylbi 205 . . . . . . . . . . . . . . 15 ((𝐹𝑥) ≠ (𝐹𝑦) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7774, 76syl6bi 241 . . . . . . . . . . . . . 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 2630 . . . . . . . . . . . . . 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 983 . . . . . . . . . . . 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 448 . . . . . . . . . . 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 449 . . . . . . . . 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 174 . . . . . . . 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 448 . . . . 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 83 . . . 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 444 . . 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 96 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 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wnel 2780  wral 2895  wrex 2896  cin 3538  wss 3539  c0 3873  {cpr 4126  cima 5031   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  0cc0 9792  1c1 9793  0cn0 11139  cuz 11519  ...cfz 12152  ..^cfzo 12289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290
This theorem is referenced by:  injresinj  12406
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