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Theorem injresinjlem 13949

Description: Lemma for injresinj 13950. (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 13932	. . . . . . 7 ⊢ ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2		fvinim0ffz 13948	. . . . . . . . . . . . 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 3066	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = 𝑌 → (𝐹‘0) = (𝐹‘𝑌))
5	4	eqcoms 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 0 → (𝐹‘0) = (𝐹‘𝑌))
6	5	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
7	6	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
8	7	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
9		ffn 6728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾))
10		1eluzge0 13031	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (ℤ_≥‘0)
11		fzoss1 13845	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^𝐾) ⊆ (0..^𝐾))
12	10, 11	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0..^𝐾))
13		fzossfz 13837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0..^𝐾) ⊆ (0...𝐾)
14	12, 13	sstrdi 3925	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0...𝐾))
15		fvelimab 6977	. . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌)))
17	16	notbid 321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌)))
18		ralnex 3092	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌))
19		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑌) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
20	19	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑋 → (¬ (𝐹‘𝑧) = (𝐹‘𝑌) ↔ ¬ (𝐹‘𝑋) = (𝐹‘𝑌)))
21	20	rspcva 3578	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑋 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌)) → ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
22		pm2.21 124	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
23	22	a1d 26	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
24	23	2a1d 27	. . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . . 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 96	. . . . . . . . . . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . . . 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 38	. . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . 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 3066	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))
37		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 = 𝑌 → (𝐹‘𝐾) = (𝐹‘𝑌))
38	37	eqcoms 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑌))
39	38	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 = 𝐾 → ((𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
40	39	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
41	40	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑌 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
42	41, 31	syl6com 38	. . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . 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 871	. . . . . . . . . . . . . 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 89	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 97	. . . . . . . . . . 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 33	. . . . . . . . . 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 102	. . . . . . . . 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 13932	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53		eqtr3 2785	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 0 ∧ 𝑌 = 0) → 𝑋 = 𝑌)
54		2a1 29	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = 𝑌 → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
55	54	2a1d 27	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → (𝐹‘0) = (𝐹‘𝑌))
58		fveq2 6905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 = 𝑋 → (𝐹‘𝐾) = (𝐹‘𝑋))
59	58	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑋))
60	59	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → (𝐹‘𝐾) = (𝐹‘𝑋))
61	57, 60	neeq12d 3020	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑌) ≠ (𝐹‘𝑋)))
62		df-ne 2960	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑌) ≠ (𝐹‘𝑋) ↔ ¬ (𝐹‘𝑌) = (𝐹‘𝑋))
63		pm2.24 125	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑌) = (𝐹‘𝑋) → (¬ (𝐹‘𝑌) = (𝐹‘𝑋) → 𝑋 = 𝑌))
64	63	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → (¬ (𝐹‘𝑌) = (𝐹‘𝑋) → 𝑋 = 𝑌))
65	64	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑌) = (𝐹‘𝑋) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
66	62, 65	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑌) ≠ (𝐹‘𝑋) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
67	61, 66	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
68	67	2a1d 27	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝑋 → (𝐹‘0) = (𝐹‘𝑋))
70	69	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 0 → (𝐹‘0) = (𝐹‘𝑋))
71	70	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘0) = (𝐹‘𝑋))
72	38	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘𝐾) = (𝐹‘𝑌))
73	71, 72	neeq12d 3020	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑋) ≠ (𝐹‘𝑌)))
74		df-ne 2960	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) ↔ ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
75	74, 22	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
76	73, 75	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
77	76	2a1d 27	. . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 𝐾) → 𝑋 = 𝑌)
79	78, 55	syl 18	. . . . . . . . . . . . 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 1053	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 18	. . . . . . . . . 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 419	. . . . . . . . 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 184	. . . . . . . 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 33	. . . . . . 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 18	. . . . . 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 418	. . . . 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 87	. . . 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 413	. . 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 33	. 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 100	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∉ wnel 3065 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 {cpr 4590 “ cima 5658 Fn wfn 6550 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 0cc0 11224 1c1 11225 ℕ₀cn0 12630 ℤ_≥cuz 12989 ...cfz 13663 ..^cfzo 13812

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-n0 12631 df-z 12718 df-uz 12990 df-fz 13664 df-fzo 13813

This theorem is used by: injresinj 13950

W3C validator