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 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) ≠ (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 12439	. . . . . . 7 ⊢ ((𝑦 ∈ (0...𝐾) ∧ ¬ 𝑦 ∈ (1..^𝐾)) → (𝑦 = 0 ∨ 𝑦 = 𝐾))
2		fvinim0ffz 12449	. . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = 𝑦 → (𝐹‘0) = (𝐹‘𝑦))
5	4	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 0 → (𝐹‘0) = (𝐹‘𝑦))
6	5	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
7	6	notbid 307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
8	7	biimpd 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
9		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾))
10		1eluzge0 11608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (ℤ_≥‘0)
11		fzoss1 12364	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^𝐾) ⊆ (0..^𝐾))
12	10, 11	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0..^𝐾))
13		fzossfz 12357	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0..^𝐾) ⊆ (0...𝐾)
14	12, 13	syl6ss 3580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0...𝐾))
15		fvelimab 6163	. . . . . . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑦)))
17	16	notbid 307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑦)))
18		ralnex 2975	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑦) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑦))
19		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
20	19	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
21	20	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑥 → (¬ (𝐹‘𝑧) = (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
22	21	rspcva 3280	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑦)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
23		pm2.21 119	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
24	23	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
25	24	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)))))
26	22, 25	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)))))
27	26	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . 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))))))
28	27	com24 93	. . . . . . . . . . . . . . . . . . . . . . 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))))))
29	18, 28	sylbir 224	. . . . . . . . . . . . . . . . . . . . . 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))))))
30	29	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))))))
31	17, 30	sylbid 229	. . . . . . . . . . . . . . . . . . . 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))))))
32	31	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))))))
33	8, 32	syl6com 36	. . . . . . . . . . . . . . . . . 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)))))))
34	3, 33	sylbi 206	. . . . . . . . . . . . . . . . 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)))))))
35	34	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)))))))
36	35	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)))))))
37		df-nel 2783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))
38		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 = 𝑦 → (𝐹‘𝐾) = (𝐹‘𝑦))
39	38	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑦))
40	39	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐾 → ((𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
41	40	notbid 307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
42	41	biimpd 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
43	42, 32	syl6com 36	. . . . . . . . . . . . . . . . . 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)))))))
44	37, 43	sylbi 206	. . . . . . . . . . . . . . . . 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)))))))
45	44	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)))))))
46	45	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)))))))
47	36, 46	jaoi 393	. . . . . . . . . . . . . 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)))))))
48	47	com13 86	. . . . . . . . . . . . 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)))))))
49	2, 48	sylbid 229	. . . . . . . . . . . 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)))))))
50	49	com14 94	. . . . . . . . . . 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)))))))
51	50	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)))))))
52	51	com15 99	. . . . . . . . 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)))))))
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) ≠ (𝐹‘𝐾) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
56	55	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)))))
57	54, 56	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)))))
58	5	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → (𝐹‘0) = (𝐹‘𝑦))
59		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 = 𝑥 → (𝐹‘𝐾) = (𝐹‘𝑥))
60	59	eqcoms 2618	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑥))
61	60	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → (𝐹‘𝐾) = (𝐹‘𝑥))
62	58, 61	neeq12d 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑥)))
63		df-ne 2782	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑦) ≠ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑦) = (𝐹‘𝑥))
64		pm2.24 120	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑦) = (𝐹‘𝑥) → (¬ (𝐹‘𝑦) = (𝐹‘𝑥) → 𝑥 = 𝑦))
65	64	eqcoms 2618	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → (¬ (𝐹‘𝑦) = (𝐹‘𝑥) → 𝑥 = 𝑦))
66	65	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑦) = (𝐹‘𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
67	63, 66	sylbi 206	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑦) ≠ (𝐹‘𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
68	62, 67	syl6bi 242	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
69	68	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)))))
70		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝑥 → (𝐹‘0) = (𝐹‘𝑥))
71	70	eqcoms 2618	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → (𝐹‘0) = (𝐹‘𝑥))
72	71	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘0) = (𝐹‘𝑥))
73	39	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘𝐾) = (𝐹‘𝑦))
74	72, 73	neeq12d 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑥) ≠ (𝐹‘𝑦)))
75		df-ne 2782	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
76	75, 23	sylbi 206	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑦) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
77	74, 76	syl6bi 242	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
78	77	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)))))
79		eqtr3 2631	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 𝐾) → 𝑥 = 𝑦)
80	79, 56	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)))))
81	57, 69, 78, 80	ccase 984	. . . . . . . . . . . 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)))))
82	81	ex 449	. . . . . . . . . . 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))))))
83	53, 82	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))))))
84	83	expcom 450	. . . . . . . . 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)))))))
85	52, 84	pm2.61i 175	. . . . . . . 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))))))
86	85	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))))))
87	1, 86	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))))))
88	87	ex 449	. . . . 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)))))))
89	88	com23 84	. . . 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)))))))
90	89	impcom 445	. . 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))))))
91	90	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))))))
92	91	com25 97	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 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

W3C validator