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Theorem injresinjlem 12450
 Description: Lemma for injresinj 12451. (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 12439 . . . . . . 7 ((𝑦 ∈ (0...𝐾) ∧ ¬ 𝑦 ∈ (1..^𝐾)) → (𝑦 = 0 ∨ 𝑦 = 𝐾))
2 fvinim0ffz 12449 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 2783 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑦 → (𝐹‘0) = (𝐹𝑦))
54eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 0 → (𝐹‘0) = (𝐹𝑦))
65eleq1d 2672 . . . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
9 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 11608 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 12364 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 12357 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13syl6ss 3580 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6163 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
169, 14, 15syl2an 493 . . . . . . . . . . . . . . . . . . . . . 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 2975 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑦) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦))
19 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
2120notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦)))
2221rspcva 3280 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑦)) → ¬ (𝐹𝑥) = (𝐹𝑦))
23 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 93 . . . . . . . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . . . 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 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)))))))
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 2783 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
38 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑦 → (𝐹𝐾) = (𝐹𝑦))
3938eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝐾 → (𝐹𝐾) = (𝐹𝑦))
4039eleq1d 2672 . . . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . . . 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 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)))))))
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 86 . . . . . . . . . . . . 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 229 . . . . . . . . . . . 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 94 . . . . . . . . . . 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 99 . . . . . . . . 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 12439 . . . . . . . . . . 11 ((𝑥 ∈ (0...𝐾) ∧ ¬ 𝑥 ∈ (1..^𝐾)) → (𝑥 = 0 ∨ 𝑥 = 𝐾))
54 eqtr3 2631 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐾𝑦 = 0) → (𝐹‘0) = (𝐹𝑦))
59 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑥 → (𝐹𝐾) = (𝐹𝑥))
6059eqcoms 2618 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐾 → (𝐹𝐾) = (𝐹𝑥))
6160adantr 480 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐾𝑦 = 0) → (𝐹𝐾) = (𝐹𝑥))
6258, 61neeq12d 2843 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐾𝑦 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑦) ≠ (𝐹𝑥)))
63 df-ne 2782 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ≠ (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
64 pm2.24 120 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) = (𝐹𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) → 𝑥 = 𝑦))
6564eqcoms 2618 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = (𝐹𝑦) → (¬ (𝐹𝑦) = (𝐹𝑥) → 𝑥 = 𝑦))
6665com12 32 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑦) = (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6763, 66sylbi 206 . . . . . . . . . . . . . . 15 ((𝐹𝑦) ≠ (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6862, 67syl6bi 242 . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . 18 (0 = 𝑥 → (𝐹‘0) = (𝐹𝑥))
7170eqcoms 2618 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝐹‘0) = (𝐹𝑥))
7271adantr 480 . . . . . . . . . . . . . . . 16 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘0) = (𝐹𝑥))
7339adantl 481 . . . . . . . . . . . . . . . 16 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹𝐾) = (𝐹𝑦))
7472, 73neeq12d 2843 . . . . . . . . . . . . . . 15 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑥) ≠ (𝐹𝑦)))
75 df-ne 2782 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) ≠ (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
7675, 23sylbi 206 . . . . . . . . . . . . . . 15 ((𝐹𝑥) ≠ (𝐹𝑦) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7774, 76syl6bi 242 . . . . . . . . . . . . . 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 2631 . . . . . . . . . . . . . 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 984 . . . . . . . . . . . 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 175 . . . . . . . 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 84 . . . 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 97 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335 This theorem is referenced by:  injresinj  12451
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