injresinjlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > injresinjlem		Structured version Visualization version GIF version

Theorem injresinjlem 13748

Description: Lemma for injresinj 13749. (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 13733	. . . . . . 7 ⊢ ((𝑌 ∈ (0...𝐾) ∧ ¬ 𝑌 ∈ (1..^𝐾)) → (𝑌 = 0 ∨ 𝑌 = 𝐾))
2		fvinim0ffz 13747	. . . . . . . . . . . . 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 3030	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = 𝑌 → (𝐹‘0) = (𝐹‘𝑌))
5	4	eqcoms 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 0 → (𝐹‘0) = (𝐹‘𝑌))
6	5	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
7	6	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
8	7	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑌 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
9		ffn 6688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾))
10		1eluzge0 12839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (ℤ_≥‘0)
11		fzoss1 13647	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^𝐾) ⊆ (0..^𝐾))
12	10, 11	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0..^𝐾))
13		fzossfz 13639	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0..^𝐾) ⊆ (0...𝐾)
14	12, 13	sstrdi 3959	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0...𝐾))
15		fvelimab 6933	. . . . . . . . . . . . . . . . . . . . . . 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 318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌)))
18		ralnex 3055	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑌) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑌))
19		fveqeq2 6867	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑌) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
20	19	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑋 → (¬ (𝐹‘𝑧) = (𝐹‘𝑌) ↔ ¬ (𝐹‘𝑋) = (𝐹‘𝑌)))
21	20	rspcva 3586	. . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 3030	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))
37		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 = 𝑌 → (𝐹‘𝐾) = (𝐹‘𝑌))
38	37	eqcoms 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑌))
39	38	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 = 𝐾 → ((𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
40	39	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑌) ∈ (𝐹 “ (1..^𝐾))))
41	40	biimpd 229	. . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 13733	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0...𝐾) ∧ ¬ 𝑋 ∈ (1..^𝐾)) → (𝑋 = 0 ∨ 𝑋 = 𝐾))
53		eqtr3 2751	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → (𝐹‘0) = (𝐹‘𝑌))
58		fveq2 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 = 𝑋 → (𝐹‘𝐾) = (𝐹‘𝑋))
59	58	eqcoms 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑋))
60	59	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → (𝐹‘𝐾) = (𝐹‘𝑋))
61	57, 60	neeq12d 2986	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = 𝐾 ∧ 𝑌 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑌) ≠ (𝐹‘𝑋)))
62		df-ne 2926	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑌) ≠ (𝐹‘𝑋) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
67	61, 66	biimtrdi 253	. . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝑋 → (𝐹‘0) = (𝐹‘𝑋))
70	69	eqcoms 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 0 → (𝐹‘0) = (𝐹‘𝑋))
71	70	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘0) = (𝐹‘𝑋))
72	38	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → (𝐹‘𝐾) = (𝐹‘𝑌))
73	71, 72	neeq12d 2986	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = 0 ∧ 𝑌 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑋) ≠ (𝐹‘𝑌)))
74		df-ne 2926	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) ↔ ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
75	74, 22	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
76	73, 75	biimtrdi 253	. . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 1037	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 413	. . . . . . . . 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 412	. . . . 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 407	. . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∉ wnel 3029 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 {cpr 4591 “ cima 5641 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 ℕ₀cn0 12442 ℤ_≥cuz 12793 ...cfz 13468 ..^cfzo 13615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616

This theorem is referenced by: injresinj 13749

W3C validator