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Theorem metucn 24475

Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24447. (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 24453	. . . . . . 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	eqtrid 2776	. . . . 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 24453	. . . . . . 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	eqtrid 2776	. . . . 5 ⊢ (𝜑 → 𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))
11	5, 10	oveq12d 7371	. . . 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 2814	. . 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 2729	. . . 4 ⊢ ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
14		eqid 2729	. . . 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 7361	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
17	16	imaeq2d 6015	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑐)))
18	17	cbvmptv 5199	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
19	18	rneqi 5883	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
20	19	metust 24462	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
21	15, 2, 20	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22		metucn.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ ∅)
23		oveq2 7361	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
24	23	imaeq2d 6015	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑑)))
25	24	cbvmptv 5199	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
26	25	rneqi 5883	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
27	26	metust 24462	. . . . 5 ⊢ ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
28	22, 7, 27	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29		oveq2 7361	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
30	29	imaeq2d 6015	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑒)))
31	30	cbvmptv 5199	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
32	31	rneqi 5883	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
33	32	metustfbas 24461	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
34	15, 2, 33	syl2anc 584	. . . 4 ⊢ (𝜑 → ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35		oveq2 7361	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
36	35	imaeq2d 6015	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑓)))
37	36	cbvmptv 5199	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
38	37	rneqi 5883	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
39	38	metustfbas 24461	. . . . 5 ⊢ ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
40	22, 7, 39	syl2anc 584	. . . 4 ⊢ (𝜑 → ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
41	13, 14, 21, 28, 34, 40	isucn2 24182	. . 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 279	. 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 2729	. . . . . . . . . 10 ⊢ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))
44		oveq2 7361	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
45	44	imaeq2d 6015	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑑 → (^◡𝐷 “ (0[,)𝑓)) = (^◡𝐷 “ (0[,)𝑑)))
46	45	rspceeqv 3602	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℝ⁺ ∧ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
47	43, 46	mpan2 691	. . . . . . . . 9 ⊢ (𝑑 ∈ ℝ⁺ → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
48	47	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
49	38	metustel 24454	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑌) → ((^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓))))
50	7, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓))))
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → ((^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓))))
52	48, 51	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))
53	26	metustel 24454	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑌) → (𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ⁺ 𝑣 = (^◡𝐷 “ (0[,)𝑑))))
54	7, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ⁺ 𝑣 = (^◡𝐷 “ (0[,)𝑑))))
55		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → 𝑣 = (^◡𝐷 “ (0[,)𝑑)))
56	55	breqd 5106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)))
57	56	imbi2d 340	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
58	57	ralbidv 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
59	58	rexralbidv 3195	. . . . . . 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))))
60	52, 54, 59	ralxfr2d 5352	. . . . . 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))))
61		eqid 2729	. . . . . . . . . . 11 ⊢ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))
62		oveq2 7361	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
63	62	imaeq2d 6015	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑐 → (^◡𝐶 “ (0[,)𝑒)) = (^◡𝐶 “ (0[,)𝑐)))
64	63	rspceeqv 3602	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ ℝ⁺ ∧ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
65	61, 64	mpan2 691	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ⁺ → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
66	65	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
67	32	metustel 24454	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (PsMet‘𝑋) → ((^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒))))
68	2, 67	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒))))
69	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → ((^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒))))
70	66, 69	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → (^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
71	19	metustel 24454	. . . . . . . . 9 ⊢ (𝐶 ∈ (PsMet‘𝑋) → (𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ⁺ 𝑢 = (^◡𝐶 “ (0[,)𝑐))))
72	2, 71	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ⁺ 𝑢 = (^◡𝐶 “ (0[,)𝑐))))
73		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → 𝑢 = (^◡𝐶 “ (0[,)𝑐)))
74	73	breqd 5106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦 ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
75	74	imbi1d 341	. . . . . . . . 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))))
76	75	2ralbidv 3193	. . . . . . . 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))))
77	70, 72, 76	rexxfr2d 5353	. . . . . . 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))))
78	77	ralbidv 3152	. . . . . 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))))
79	60, 78	bitrd 279	. . . . 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))))
80	79	adantr 480	. . . 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))))
81	2	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
82		simplr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑐 ∈ ℝ⁺)
83		simprr 772	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
84		simprl 770	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
85		elbl4 24467	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
86		rpxr 12921	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
87		elbl3ps 24295	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ^*) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
88	86, 87	sylanl2 681	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
89	85, 88	bitr3d 281	. . . . . . . . 9 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
90	81, 82, 83, 84, 89	syl22anc 838	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
91	7	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
92		simpllr 775	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑑 ∈ ℝ⁺)
93		simp-4r 783	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
94	93, 83	ffvelcdmd 7023	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
95	93, 84	ffvelcdmd 7023	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
96		elbl4 24467	. . . . . . . . . 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)))
97		rpxr 12921	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ^*)
98		elbl3ps 24295	. . . . . . . . . . 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))
99	97, 98	sylanl2 681	. . . . . . . . . 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))
100	96, 99	bitr3d 281	. . . . . . . . 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))
101	91, 92, 94, 95, 100	syl22anc 838	. . . . . . . 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))
102	90, 101	imbi12d 344	. . . . . . 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)))
103	102	2ralbidva 3191	. . . . . 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)))
104	103	rexbidva 3151	. . . . 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)))
105	104	ralbidva 3150	. . . 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)))
106	80, 105	bitrd 279	. . 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)))
107	106	pm5.32da 579	. 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))))
108	42, 107	bitrd 279	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∅c0 4286 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 ^◡ccnv 5622 ran crn 5624 “ cima 5626 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 0cc0 11028 ℝ^*cxr 11167 < clt 11168 ℝ⁺crp 12911 [,)cico 13268 PsMetcpsmet 21263 ballcbl 21266 fBascfbas 21267 filGencfg 21268 metUnifcmetu 21270 UnifOncust 24103 Cn_ucucn 24178

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ico 13272 df-psmet 21271 df-bl 21274 df-fbas 21276 df-fg 21277 df-metu 21278 df-fil 23749 df-ust 24104 df-ucn 24179

This theorem is referenced by: qqhucn 33958 heicant 37634

W3C validator