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Theorem injresinjlem 12544
Description: Lemma for injresinj 12545. (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 12530 . . . . . . 7 ((𝑦 ∈ (0...𝐾) ∧ ¬ 𝑦 ∈ (1..^𝐾)) → (𝑦 = 0 ∨ 𝑦 = 𝐾))
2 fvinim0ffz 12543 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3 df-nel 2894 . . . . . . . . . . . . . . . . . 18 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 𝑦 → (𝐹‘0) = (𝐹𝑦))
54eqcoms 2629 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 0 → (𝐹‘0) = (𝐹𝑦))
65eleq1d 2683 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
76notbid 308 . . . . . . . . . . . . . . . . . . . 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 6012 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
10 1eluzge0 11692 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (ℤ‘0)
11 fzoss1 12452 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘0) → (1..^𝐾) ⊆ (0..^𝐾))
1210, 11mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0..^𝐾))
13 fzossfz 12445 . . . . . . . . . . . . . . . . . . . . . . . 24 (0..^𝐾) ⊆ (0...𝐾)
1412, 13syl6ss 3600 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 ∈ ℕ0 → (1..^𝐾) ⊆ (0...𝐾))
15 fvelimab 6220 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
169, 14, 15syl2an 494 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
1716notbid 308 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (¬ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦)))
18 ralnex 2988 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹𝑧) = (𝐹𝑦) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹𝑧) = (𝐹𝑦))
19 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
2120notbid 308 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑥 → (¬ (𝐹𝑧) = (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦)))
2221rspcva 3297 . . . . . . . . . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . . . . . . . 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 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)))))))
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 2894 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
38 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐾 = 𝑦 → (𝐹𝐾) = (𝐹𝑦))
3938eqcoms 2629 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝐾 → (𝐹𝐾) = (𝐹𝑦))
4039eleq1d 2683 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐾 → ((𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝑦) ∈ (𝐹 “ (1..^𝐾))))
4140notbid 308 . . . . . . . . . . . . . . . . . . . 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 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)))))))
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 394 . . . . . . . . . . . . . 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 12530 . . . . . . . . . . 11 ((𝑥 ∈ (0...𝐾) ∧ ¬ 𝑥 ∈ (1..^𝐾)) → (𝑥 = 0 ∨ 𝑥 = 𝐾))
54 eqtr3 2642 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐾𝑦 = 0) → (𝐹‘0) = (𝐹𝑦))
59 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝐾 = 𝑥 → (𝐹𝐾) = (𝐹𝑥))
6059eqcoms 2629 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐾 → (𝐹𝐾) = (𝐹𝑥))
6160adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝐾𝑦 = 0) → (𝐹𝐾) = (𝐹𝑥))
6258, 61neeq12d 2851 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐾𝑦 = 0) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑦) ≠ (𝐹𝑥)))
63 df-ne 2791 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ≠ (𝐹𝑥) ↔ ¬ (𝐹𝑦) = (𝐹𝑥))
64 pm2.24 121 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) = (𝐹𝑥) → (¬ (𝐹𝑦) = (𝐹𝑥) → 𝑥 = 𝑦))
6564eqcoms 2629 . . . . . . . . . . . . . . . . 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 6158 . . . . . . . . . . . . . . . . . 18 (0 = 𝑥 → (𝐹‘0) = (𝐹𝑥))
7170eqcoms 2629 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝐹‘0) = (𝐹𝑥))
7271adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘0) = (𝐹𝑥))
7339adantl 482 . . . . . . . . . . . . . . . 16 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹𝐾) = (𝐹𝑦))
7472, 73neeq12d 2851 . . . . . . . . . . . . . . 15 ((𝑥 = 0 ∧ 𝑦 = 𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) ↔ (𝐹𝑥) ≠ (𝐹𝑦)))
75 df-ne 2791 . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . 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 986 . . . . . . . . . . . 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 450 . . . . . . . . . . 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 451 . . . . . . . . 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 450 . . . . 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 446 . . 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 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wnel 2893  wral 2908  wrex 2909  cin 3559  wss 3560  c0 3897  {cpr 4157  cima 5087   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  0cc0 9896  1c1 9897  0cn0 11252  cuz 11647  ...cfz 12284  ..^cfzo 12422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423
This theorem is referenced by:  injresinj  12545
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