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Theorem metucn 22376

Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 22348. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)

**Hypotheses**
Ref	Expression
metucn.u	⊢ 𝑈 = (metUnif‘𝐶)
metucn.v	⊢ 𝑉 = (metUnif‘𝐷)
metucn.x	⊢ (𝜑 → 𝑋 ≠ ∅)
metucn.y	⊢ (𝜑 → 𝑌 ≠ ∅)
metucn.c	⊢ (𝜑 → 𝐶 ∈ (PsMet‘𝑋))
metucn.d	⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑌))

**Assertion**
Ref	Expression
metucn	⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))))

Distinct variable groups: 𝑐,𝑑,𝑥,𝑦,𝐶 𝐷,𝑐,𝑑,𝑥,𝑦 𝐹,𝑐,𝑑,𝑥,𝑦 𝑥,𝑈,𝑦 𝑥,𝑉 𝑋,𝑐,𝑑,𝑥,𝑦 𝑌,𝑐,𝑑,𝑥,𝑦 𝜑,𝑐,𝑑,𝑥,𝑦 Allowed substitution hints: 𝑈(𝑐,𝑑) 𝑉(𝑦,𝑐,𝑑)

Proof of Theorem metucn Dummy variables 𝑎 𝑒 𝑢 𝑣 𝑏 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metucn.u	. . . . . 6 ⊢ 𝑈 = (metUnif‘𝐶)
2		metucn.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (PsMet‘𝑋))
3		metuval 22354	. . . . . . 7 ⊢ (𝐶 ∈ (PsMet‘𝑋) → (metUnif‘𝐶) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → (metUnif‘𝐶) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))))
5	1, 4	syl5eq 2668	. . . . 5 ⊢ (𝜑 → 𝑈 = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))))
6		metucn.v	. . . . . 6 ⊢ 𝑉 = (metUnif‘𝐷)
7		metucn.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑌))
8		metuval 22354	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑌) → (metUnif‘𝐷) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (metUnif‘𝐷) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))
10	6, 9	syl5eq 2668	. . . . 5 ⊢ (𝜑 → 𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))
11	5, 10	oveq12d 6668	. . . 4 ⊢ (𝜑 → (𝑈 Cn_u𝑉) = (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) Cn_u((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))))
12	11	eleq2d 2687	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ 𝐹 ∈ (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) Cn_u((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))))
13		eqid 2622	. . . 4 ⊢ ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
14		eqid 2622	. . . 4 ⊢ ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))
15		metucn.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
16		oveq2 6658	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
17	16	imaeq2d 5466	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑐)))
18	17	cbvmptv 4750	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
19	18	rneqi 5352	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
20	19	metust 22363	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
21	15, 2, 20	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22		metucn.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ ∅)
23		oveq2 6658	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
24	23	imaeq2d 5466	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑑)))
25	24	cbvmptv 4750	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
26	25	rneqi 5352	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
27	26	metust 22363	. . . . 5 ⊢ ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
28	22, 7, 27	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29		oveq2 6658	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
30	29	imaeq2d 5466	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑒)))
31	30	cbvmptv 4750	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
32	31	rneqi 5352	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
33	32	metustfbas 22362	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
34	15, 2, 33	syl2anc 693	. . . 4 ⊢ (𝜑 → ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35		oveq2 6658	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
36	35	imaeq2d 5466	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑓)))
37	36	cbvmptv 4750	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
38	37	rneqi 5352	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
39	38	metustfbas 22362	. . . . 5 ⊢ ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
40	22, 7, 39	syl2anc 693	. . . 4 ⊢ (𝜑 → ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
41	13, 14, 21, 28, 34, 40	isucn2 22083	. . 3 ⊢ (𝜑 → (𝐹 ∈ (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) Cn_u((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))))
42	12, 41	bitrd 268	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))))
43		eqid 2622	. . . . . . . . . 10 ⊢ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))
44		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
45	44	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → (^◡𝐷 “ (0[,)𝑓)) = (^◡𝐷 “ (0[,)𝑑)))
46	45	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑑 → ((^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)) ↔ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))))
47	46	rspcev 3309	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℝ⁺ ∧ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
48	43, 47	mpan2 707	. . . . . . . . 9 ⊢ (𝑑 ∈ ℝ⁺ → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
49	48	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
50	38	metustel 22355	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑌) → ((^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓))))
51	7, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓))))
52	51	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → ((^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓))))
53	49, 52	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))
54	26	metustel 22355	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑌) → (𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ⁺ 𝑣 = (^◡𝐷 “ (0[,)𝑑))))
55	7, 54	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ⁺ 𝑣 = (^◡𝐷 “ (0[,)𝑑))))
56		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → 𝑣 = (^◡𝐷 “ (0[,)𝑑)))
57	56	breqd 4664	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)))
58	57	imbi2d 330	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
59	58	ralbidv 2986	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
60	59	rexralbidv 3058	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → (∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
61	53, 55, 60	ralxfr2d 4882	. . . . . 6 ⊢ (𝜑 → (∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
62		eqid 2622	. . . . . . . . . . 11 ⊢ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))
63		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
64	63	imaeq2d 5466	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑐 → (^◡𝐶 “ (0[,)𝑒)) = (^◡𝐶 “ (0[,)𝑐)))
65	64	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑐 → ((^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)) ↔ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))))
66	65	rspcev 3309	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ ℝ⁺ ∧ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
67	62, 66	mpan2 707	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ⁺ → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
68	67	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
69	32	metustel 22355	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (PsMet‘𝑋) → ((^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒))))
70	2, 69	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒))))
71	70	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → ((^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒))))
72	68, 71	mpbird 247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → (^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
73	19	metustel 22355	. . . . . . . . 9 ⊢ (𝐶 ∈ (PsMet‘𝑋) → (𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ⁺ 𝑢 = (^◡𝐶 “ (0[,)𝑐))))
74	2, 73	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ⁺ 𝑢 = (^◡𝐶 “ (0[,)𝑐))))
75		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → 𝑢 = (^◡𝐶 “ (0[,)𝑐)))
76	75	breqd 4664	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦 ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
77	76	imbi1d 331	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
78	77	2ralbidv 2989	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
79	72, 74, 78	rexxfr2d 4883	. . . . . . 7 ⊢ (𝜑 → (∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
80	79	ralbidv 2986	. . . . . 6 ⊢ (𝜑 → (∀𝑑 ∈ ℝ⁺ ∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
81	61, 80	bitrd 268	. . . . 5 ⊢ (𝜑 → (∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
82	81	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
83	2	ad4antr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
84		simplr 792	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑐 ∈ ℝ⁺)
85		simprr 796	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
86		simprl 794	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
87		elbl4 22368	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
88		rpxr 11840	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
89		elbl3ps 22196	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ^*) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
90	88, 89	sylanl2 683	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
91	87, 90	bitr3d 270	. . . . . . . . 9 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
92	83, 84, 85, 86, 91	syl22anc 1327	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
93	7	ad4antr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
94		simpllr 799	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑑 ∈ ℝ⁺)
95		simp-4r 807	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
96	95, 85	ffvelrnd 6360	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
97	95, 86	ffvelrnd 6360	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
98		elbl4 22368	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ ((𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑥) ∈ 𝑌)) → ((𝐹‘𝑥) ∈ ((𝐹‘𝑦)(ball‘𝐷)𝑑) ↔ (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)))
99		rpxr 11840	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ^*)
100		elbl3ps 22196	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ^*) ∧ ((𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑥) ∈ 𝑌)) → ((𝐹‘𝑥) ∈ ((𝐹‘𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))
101	99, 100	sylanl2 683	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ ((𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑥) ∈ 𝑌)) → ((𝐹‘𝑥) ∈ ((𝐹‘𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))
102	98, 101	bitr3d 270	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ ((𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑥) ∈ 𝑌)) → ((𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦) ↔ ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))
103	93, 94, 96, 97, 102	syl22anc 1327	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦) ↔ ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))
104	92, 103	imbi12d 334	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑)))
105	104	2ralbidva 2988	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑)))
106	105	rexbidva 3049	. . . . 5 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) → (∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑)))
107	106	ralbidva 2985	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑)))
108	82, 107	bitrd 268	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑)))
109	108	pm5.32da 673	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))))
110	42, 109	bitrd 268	1 ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∅c0 3915 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 ran crn 5115 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℝ^*cxr 10073 < clt 10074 ℝ⁺crp 11832 [,)cico 12177 PsMetcpsmet 19730 ballcbl 19733 fBascfbas 19734 filGencfg 19735 metUnifcmetu 19737 UnifOncust 22003 Cn_ucucn 22079

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-psmet 19738 df-bl 19741 df-fbas 19743 df-fg 19744 df-metu 19745 df-fil 21650 df-ust 22004 df-ucn 22080

This theorem is referenced by: qqhucn 30036 heicant 33444

W3C validator