bfp - Mathbox for Jeff Madsen

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Theorem bfp 36692

Description: Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if 𝐹 has two fixed points, then the distance between them is less than 𝐾 times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

**Hypotheses**
Ref	Expression
bfp.2	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bfp.3	⊢ (𝜑 → 𝑋 ≠ ∅)
bfp.4	⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
bfp.5	⊢ (𝜑 → 𝐾 < 1)
bfp.6	⊢ (𝜑 → 𝐹:𝑋⟶𝑋)
bfp.7	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))

**Assertion**
Ref	Expression
bfp	⊢ (𝜑 → ∃!𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧)

Distinct variable groups: 𝑥,𝑦,𝑧,𝐷 𝜑,𝑥,𝑦 𝑥,𝐹,𝑦,𝑧 𝑥,𝐾,𝑦 𝑥,𝑋,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑧) 𝐾(𝑧)

Proof of Theorem bfp Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bfp.3	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
2		n0 4347	. . . 4 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑋)
3	1, 2	sylib 217	. . 3 ⊢ (𝜑 → ∃𝑤 𝑤 ∈ 𝑋)
4		bfp.2	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
5	4	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝐷 ∈ (CMet‘𝑋))
6	1	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑋 ≠ ∅)
7		bfp.4	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
8	7	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝐾 ∈ ℝ⁺)
9		bfp.5	. . . . 5 ⊢ (𝜑 → 𝐾 < 1)
10	9	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝐾 < 1)
11		bfp.6	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶𝑋)
12	11	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝐹:𝑋⟶𝑋)
13		bfp.7	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
14	13	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
15		eqid 2733	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
16		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ 𝑋)
17		eqid 2733	. . . 4 ⊢ seq1((𝐹 ∘ 1^st ), (ℕ × {𝑤})) = seq1((𝐹 ∘ 1^st ), (ℕ × {𝑤}))
18	5, 6, 8, 10, 12, 14, 15, 16, 17	bfplem2 36691	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧)
19	3, 18	exlimddv 1939	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧)
20		oveq12 7418	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) = (𝑥𝐷𝑦))
21	20	adantl 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) = (𝑥𝐷𝑦))
22	13	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
23	21, 22	eqbrtrrd 5173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (𝑥𝐷𝑦) ≤ (𝐾 · (𝑥𝐷𝑦)))
24		cmetmet 24803	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
25	4, 24	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
26	25	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 𝐷 ∈ (Met‘𝑋))
27		simplrl 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 𝑥 ∈ 𝑋)
28		simplrr 777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑋)
29		metcl 23838	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ)
30	26, 27, 28, 29	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (𝑥𝐷𝑦) ∈ ℝ)
31	7	rpred 13016	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ)
32	31	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 𝐾 ∈ ℝ)
33	32, 30	remulcld 11244	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (𝐾 · (𝑥𝐷𝑦)) ∈ ℝ)
34	30, 33	suble0d 11805	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (((𝑥𝐷𝑦) − (𝐾 · (𝑥𝐷𝑦))) ≤ 0 ↔ (𝑥𝐷𝑦) ≤ (𝐾 · (𝑥𝐷𝑦))))
35	23, 34	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((𝑥𝐷𝑦) − (𝐾 · (𝑥𝐷𝑦))) ≤ 0)
36		1cnd 11209	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 1 ∈ ℂ)
37	32	recnd 11242	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 𝐾 ∈ ℂ)
38	30	recnd 11242	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (𝑥𝐷𝑦) ∈ ℂ)
39	36, 37, 38	subdird 11671	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((1 − 𝐾) · (𝑥𝐷𝑦)) = ((1 · (𝑥𝐷𝑦)) − (𝐾 · (𝑥𝐷𝑦))))
40	38	mullidd 11232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (1 · (𝑥𝐷𝑦)) = (𝑥𝐷𝑦))
41	40	oveq1d 7424	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((1 · (𝑥𝐷𝑦)) − (𝐾 · (𝑥𝐷𝑦))) = ((𝑥𝐷𝑦) − (𝐾 · (𝑥𝐷𝑦))))
42	39, 41	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((1 − 𝐾) · (𝑥𝐷𝑦)) = ((𝑥𝐷𝑦) − (𝐾 · (𝑥𝐷𝑦))))
43		1re 11214	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
44		resubcl 11524	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 − 𝐾) ∈ ℝ)
45	43, 31, 44	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 − 𝐾) ∈ ℝ)
46	45	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (1 − 𝐾) ∈ ℝ)
47	46	recnd 11242	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (1 − 𝐾) ∈ ℂ)
48	47	mul01d 11413	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((1 − 𝐾) · 0) = 0)
49	35, 42, 48	3brtr4d 5181	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((1 − 𝐾) · (𝑥𝐷𝑦)) ≤ ((1 − 𝐾) · 0))
50		0red 11217	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 0 ∈ ℝ)
51		posdif 11707	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐾 < 1 ↔ 0 < (1 − 𝐾)))
52	31, 43, 51	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 < 1 ↔ 0 < (1 − 𝐾)))
53	9, 52	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 − 𝐾))
54	53	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 0 < (1 − 𝐾))
55		lemul2 12067	. . . . . . . . 9 ⊢ (((𝑥𝐷𝑦) ∈ ℝ ∧ 0 ∈ ℝ ∧ ((1 − 𝐾) ∈ ℝ ∧ 0 < (1 − 𝐾))) → ((𝑥𝐷𝑦) ≤ 0 ↔ ((1 − 𝐾) · (𝑥𝐷𝑦)) ≤ ((1 − 𝐾) · 0)))
56	30, 50, 46, 54, 55	syl112anc 1375	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((𝑥𝐷𝑦) ≤ 0 ↔ ((1 − 𝐾) · (𝑥𝐷𝑦)) ≤ ((1 − 𝐾) · 0)))
57	49, 56	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (𝑥𝐷𝑦) ≤ 0)
58		metge0 23851	. . . . . . . 8 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦))
59	26, 27, 28, 58	syl3anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 0 ≤ (𝑥𝐷𝑦))
60		0re 11216	. . . . . . . 8 ⊢ 0 ∈ ℝ
61		letri3 11299	. . . . . . . 8 ⊢ (((𝑥𝐷𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
62	30, 60, 61	sylancl 587	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
63	57, 59, 62	mpbir2and 712	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → (𝑥𝐷𝑦) = 0)
64		meteq0 23845	. . . . . . 7 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
65	26, 27, 28, 64	syl3anc 1372	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
66	63, 65	mpbid 231	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ ((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦)) → 𝑥 = 𝑦)
67	66	ex 414	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) → 𝑥 = 𝑦))
68	67	ralrimivva 3201	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) → 𝑥 = 𝑦))
69		fveq2 6892	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
70		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
71	69, 70	eqeq12d 2749	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑧) = 𝑧))
72	71	anbi1d 631	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) ↔ ((𝐹‘𝑧) = 𝑧 ∧ (𝐹‘𝑦) = 𝑦)))
73		equequ1 2029	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
74	72, 73	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑧 → ((((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) → 𝑥 = 𝑦) ↔ (((𝐹‘𝑧) = 𝑧 ∧ (𝐹‘𝑦) = 𝑦) → 𝑧 = 𝑦)))
75	74	ralbidv 3178	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑋 (((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝑋 (((𝐹‘𝑧) = 𝑧 ∧ (𝐹‘𝑦) = 𝑦) → 𝑧 = 𝑦)))
76	75	cbvralvw 3235	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝐹‘𝑥) = 𝑥 ∧ (𝐹‘𝑦) = 𝑦) → 𝑥 = 𝑦) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝐹‘𝑧) = 𝑧 ∧ (𝐹‘𝑦) = 𝑦) → 𝑧 = 𝑦))
77	68, 76	sylib 217	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝐹‘𝑧) = 𝑧 ∧ (𝐹‘𝑦) = 𝑦) → 𝑧 = 𝑦))
78		fveq2 6892	. . . 4 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
79		id 22	. . . 4 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
80	78, 79	eqeq12d 2749	. . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = 𝑧 ↔ (𝐹‘𝑦) = 𝑦))
81	80	reu4 3728	. 2 ⊢ (∃!𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧 ↔ (∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧 ∧ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝐹‘𝑧) = 𝑧 ∧ (𝐹‘𝑦) = 𝑦) → 𝑧 = 𝑦)))
82	19, 77, 81	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 ∅c0 4323 {csn 4629 class class class wbr 5149 × cxp 5675 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 ℝcr 11109 0cc0 11110 1c1 11111 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℝ⁺crp 12974 seqcseq 13966 Metcmet 20930 MetOpencmopn 20934 CMetccmet 24771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-rest 17368 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-top 22396 df-topon 22413 df-bases 22449 df-ntr 22524 df-nei 22602 df-lm 22733 df-haus 22819 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-cfil 24772 df-cau 24773 df-cmet 24774

This theorem is referenced by: (None)

W3C validator