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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) ≠ (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 12530	. . . . . . 7 ⊢ ((𝑦 ∈ (0...𝐾) ∧ ¬ 𝑦 ∈ (1..^𝐾)) → (𝑦 = 0 ∨ 𝑦 = 𝐾))
2		fvinim0ffz 12543	. . . . . . . . . . . . 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 2894	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
4		fveq2 6158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = 𝑦 → (𝐹‘0) = (𝐹‘𝑦))
5	4	eqcoms 2629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 0 → (𝐹‘0) = (𝐹‘𝑦))
6	5	eleq1d 2683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → ((𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
7	6	notbid 308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
8	7	biimpd 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 0 → (¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
9		ffn 6012	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾))
10		1eluzge0 11692	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (ℤ_≥‘0)
11		fzoss1 12452	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^𝐾) ⊆ (0..^𝐾))
12	10, 11	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0..^𝐾))
13		fzossfz 12445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0..^𝐾) ⊆ (0...𝐾)
14	12, 13	syl6ss 3600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ ℕ₀ → (1..^𝐾) ⊆ (0...𝐾))
15		fvelimab 6220	. . . . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑦)))
17	16	notbid 308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑦)))
18		ralnex 2988	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑦) ↔ ¬ ∃𝑧 ∈ (1..^𝐾)(𝐹‘𝑧) = (𝐹‘𝑦))
19		fveq2 6158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
20	19	eqeq1d 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
21	20	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑥 → (¬ (𝐹‘𝑧) = (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
22	21	rspcva 3297	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (1..^𝐾) ∧ ∀𝑧 ∈ (1..^𝐾) ¬ (𝐹‘𝑧) = (𝐹‘𝑦)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
23		pm2.21 120	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . . . . . . . . . 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 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))))))
29	18, 28	sylbir 225	. . . . . . . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . . . . . 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 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)))))))
34	3, 33	sylbi 207	. . . . . . . . . . . . . . . . 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 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)))))))
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 2894	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))
38		fveq2 6158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 = 𝑦 → (𝐹‘𝐾) = (𝐹‘𝑦))
39	38	eqcoms 2629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑦))
40	39	eleq1d 2683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐾 → ((𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
41	40	notbid 308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
42	41	biimpd 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐾 → (¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ (1..^𝐾))))
43	42, 32	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)))))))
44	37, 43	sylbi 207	. . . . . . . . . . . . . . . . 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 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)))))))
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 394	. . . . . . . . . . . . . 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 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)))))))
49	2, 48	sylbid 230	. . . . . . . . . . . 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 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)))))))
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 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)))))))
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) ≠ (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 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → (𝐹‘0) = (𝐹‘𝑦))
59		fveq2 6158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 = 𝑥 → (𝐹‘𝐾) = (𝐹‘𝑥))
60	59	eqcoms 2629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐾 → (𝐹‘𝐾) = (𝐹‘𝑥))
61	60	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → (𝐹‘𝐾) = (𝐹‘𝑥))
62	58, 61	neeq12d 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐾 ∧ 𝑦 = 0) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑥)))
63		df-ne 2791	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑦) ≠ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑦) = (𝐹‘𝑥))
64		pm2.24 121	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑦) = (𝐹‘𝑥) → (¬ (𝐹‘𝑦) = (𝐹‘𝑥) → 𝑥 = 𝑦))
65	64	eqcoms 2629	. . . . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑦) ≠ (𝐹‘𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
68	62, 67	syl6bi 243	. . . . . . . . . . . . . 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 6158	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝑥 → (𝐹‘0) = (𝐹‘𝑥))
71	70	eqcoms 2629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → (𝐹‘0) = (𝐹‘𝑥))
72	71	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘0) = (𝐹‘𝑥))
73	39	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → (𝐹‘𝐾) = (𝐹‘𝑦))
74	72, 73	neeq12d 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) ↔ (𝐹‘𝑥) ≠ (𝐹‘𝑦)))
75		df-ne 2791	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
76	75, 23	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑦) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
77	74, 76	syl6bi 243	. . . . . . . . . . . . . 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 2642	. . . . . . . . . . . . . 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 986	. . . . . . . . . . . 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 450	. . . . . . . . . . 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 451	. . . . . . . . 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 176	. . . . . . . 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 450	. . . . 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 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)))))))
90	89	impcom 446	. . 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 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 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 ℕ₀cn0 11252 ℤ_≥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

W3C validator