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Theorem injresinjlem 13692

Description: Lemma for injresinj 13693. (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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))

Proof of Theorem injresinjlem Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznelfzo 13677	. . . . . . 7 ⊢ ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2		fvinim0ffz 13691	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3		df-nel 3050	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = 𝑌 → (𝐹‘0) = (𝐹‘𝑌))
5	4	eqcoms 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 0 → (𝐹‘0) = (𝐹‘𝑌))
6	5	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
7	6	notbid 317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
8	7	biimpd 228	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
9		ffn 6668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾))
10		1eluzge0 12817	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (ℤ_≥‘0)
11		fzoss1 13599	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^𝐾) ⊆ (0..^𝐾))
12	10, 11	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0..^𝐾))
13		fzossfz 13591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0..^𝐾) ⊆ (0...𝐾)
14	12, 13	sstrdi 3956	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0...𝐾))
15		fvelimab 6914	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 Fn (0...𝐾) ∧ (1..^𝐾) ⊆ (0...𝐾)) → ((𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌)))
16	9, 14, 15	syl2an 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌)))
17	16	notbid 317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌)))
18		ralnex 3075	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌))
19		fveqeq2 6851	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑌) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
20	19	notbid 317	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑋 → (¬ (𝐹‘𝑧) = (𝐹‘𝑌) ↔ ¬ (𝐹‘𝑋) = (𝐹‘𝑌)))
21	20	rspcva 3579	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑋 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌)) → ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
22		pm2.21 123	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
23	22	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
24	23	2a1d 26	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ (𝐹‘𝑋) = (𝐹‘𝑌) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
25	21, 24	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌)) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
26	25	expcom 414	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌) → (𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
27	26	com24 95	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
28	18, 27	sylbir 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
29	28	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
30	17, 29	sylbid 239	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
31	30	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
32	8, 31	syl6com 37	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → (𝑌 = 0 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
33	3, 32	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) → (𝑌 = 0 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
34	33	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))) → (𝑌 = 0 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
35	34	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = 0 → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
36		df-nel 3050	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))
37		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 = 𝑌 → (𝐹‘𝐾) = (𝐹‘𝑌))
38	37	eqcoms 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑌))
39	38	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 = 𝐾 → ((𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
40	39	notbid 317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
41	40	biimpd 228	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑌 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
42	41, 31	syl6com 37	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) → (𝑌 = 𝐾 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
43	36, 42	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) → (𝑌 = 𝐾 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
44	43	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))) → (𝑌 = 𝐾 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
45	44	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = 𝐾 → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
46	35, 45	jaoi 855	. . . . . . . . . . . . . 14 ⊢ ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
47	46	com13 88	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
48	2, 47	sylbid 239	. . . . . . . . . . . 12 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (𝑋 ∈ (0...𝐾) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
49	48	com14 96	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (0...𝐾) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
50	49	com12 32	. . . . . . . . . 10 ⊢ (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (𝑋 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
51	50	com15 101	. . . . . . . . 9 ⊢ (𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
52		elfznelfzo 13677	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53		eqtr3 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 0 ∧ 𝑌 = 0) → 𝑋 = 𝑌)
54		2a1 28	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = 𝑌 → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
55	54	2a1d 26	. . . . . . . . . . . . . 14 ⊢ (𝑋 = 𝑌 → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
56	53, 55	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑋 = 0 ∧ 𝑌 = 0) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
57	5	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → (𝐹‘0) = (𝐹‘𝑌))
58		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 = 𝑋 → (𝐹‘𝐾) = (𝐹‘𝑋))
59	58	eqcoms 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑋))
60	59	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → (𝐹‘𝐾) = (𝐹‘𝑋))
61	57, 60	neeq12d 3005	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑌) ≠ (𝐹‘𝑋)))
62		df-ne 2944	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑌) ≠ (𝐹‘𝑋) ↔ ¬ (𝐹‘𝑌) = (𝐹‘𝑋))
63		pm2.24 124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑌) = (𝐹‘𝑋) → (¬ (𝐹‘𝑌) = (𝐹‘𝑋) → 𝑋 = 𝑌))
64	63	eqcoms 2744	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → (¬ (𝐹‘𝑌) = (𝐹‘𝑋) → 𝑋 = 𝑌))
65	64	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑌) = (𝐹‘𝑋) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
66	62, 65	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑌) ≠ (𝐹‘𝑋) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
67	61, 66	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
68	67	2a1d 26	. . . . . . . . . . . . 13 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
69		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝑋 → (𝐹‘0) = (𝐹‘𝑋))
70	69	eqcoms 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 0 → (𝐹‘0) = (𝐹‘𝑋))
71	70	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘0) = (𝐹‘𝑋))
72	38	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘𝐾) = (𝐹‘𝑌))
73	71, 72	neeq12d 3005	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑋) ≠ (𝐹‘𝑌)))
74		df-ne 2944	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) ↔ ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
75	74, 22	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
76	73, 75	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
77	76	2a1d 26	. . . . . . . . . . . . 13 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
78		eqtr3 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 𝐾) → 𝑋 = 𝑌)
79	78, 55	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
80	56, 68, 77, 79	ccase 1036	. . . . . . . . . . . 12 ⊢ (((𝑋 = 0 ∨ 𝑋 = 𝐾) ∧ (𝑌 = 0 ∨ 𝑌 = 𝐾)) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))
81	80	ex 413	. . . . . . . . . . 11 ⊢ ((𝑋 = 0 ∨ 𝑋 = 𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
82	52, 81	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
83	82	expcom 414	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
84	51, 83	pm2.61i 182	. . . . . . . 8 ⊢ (𝑋 ∈ (0...𝐾) → ((𝑌 = 0 ∨ 𝑌 = 𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
85	84	com12 32	. . . . . . 7 ⊢ ((𝑌 = 0 ∨ 𝑌 = 𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
86	1, 85	syl 17	. . . . . 6 ⊢ ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
87	86	ex 413	. . . . 5 ⊢ (𝑌 ∈ (0...𝐾) → (¬ 𝑌 ∈ (1..^𝐾) → (𝑋 ∈ (0...𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
88	87	com23 86	. . . 4 ⊢ (𝑌 ∈ (0...𝐾) → (𝑋 ∈ (0...𝐾) → (¬ 𝑌 ∈ (1..^𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))))
89	88	impcom 408	. . 3 ⊢ ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → (¬ 𝑌 ∈ (1..^𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
90	89	com12 32	. 2 ⊢ (¬ 𝑌 ∈ (1..^𝐾) → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
91	90	com25 99	1 ⊢ (¬ 𝑌 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∉ wnel 3049 ∀wral 3064 ∃wrex 3073 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 {cpr 4588 “ cima 5636 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 ℕ₀cn0 12413 ℤ_≥cuz 12763 ...cfz 13424 ..^cfzo 13567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568

This theorem is referenced by: injresinj 13693

W3C validator