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Theorem metucn 24638

Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24610. (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 24616	. . . . . . 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 2810	. . . . 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 24616	. . . . . . 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 2810	. . . . 5 ⊢ (𝜑 → 𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))
11	5, 10	oveq12d 7414	. . . 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 2849	. . 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 2763	. . . 4 ⊢ ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
14		eqid 2763	. . . 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 7404	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
17	16	imaeq2d 6049	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑐)))
18	17	cbvmptv 5205	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
19	18	rneqi 5914	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
20	19	metust 24625	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
21	15, 2, 20	syl2anc 593	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22		metucn.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ ∅)
23		oveq2 7404	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
24	23	imaeq2d 6049	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑑)))
25	24	cbvmptv 5205	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
26	25	rneqi 5914	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
27	26	metust 24625	. . . . 5 ⊢ ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
28	22, 7, 27	syl2anc 593	. . . 4 ⊢ (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29		oveq2 7404	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
30	29	imaeq2d 6049	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑒)))
31	30	cbvmptv 5205	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
32	31	rneqi 5914	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
33	32	metustfbas 24624	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
34	15, 2, 33	syl2anc 593	. . . 4 ⊢ (𝜑 → ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35		oveq2 7404	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
36	35	imaeq2d 6049	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑓)))
37	36	cbvmptv 5205	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
38	37	rneqi 5914	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
39	38	metustfbas 24624	. . . . 5 ⊢ ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
40	22, 7, 39	syl2anc 593	. . . 4 ⊢ (𝜑 → ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
41	13, 14, 21, 28, 34, 40	isucn2 24345	. . 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 281	. 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 2763	. . . . . . . . . 10 ⊢ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))
44		oveq2 7404	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
45	44	imaeq2d 6049	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑑 → (^◡𝐷 “ (0[,)𝑓)) = (^◡𝐷 “ (0[,)𝑑)))
46	45	rspceeqv 3605	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℝ⁺ ∧ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
47	43, 46	mpan2 701	. . . . . . . . 9 ⊢ (𝑑 ∈ ℝ⁺ → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
48	47	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
49	38	metustel 24617	. . . . . . . . . 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 484	. . . . . . . 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 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))
53	26	metustel 24617	. . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → 𝑣 = (^◡𝐷 “ (0[,)𝑑)))
56	55	breqd 5112	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)))
57	56	imbi2d 342	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
58	57	ralbidv 3186	. . . . . . . 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 3229	. . . . . . 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 5368	. . . . . 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 2763	. . . . . . . . . . 11 ⊢ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))
62		oveq2 7404	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
63	62	imaeq2d 6049	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑐 → (^◡𝐶 “ (0[,)𝑒)) = (^◡𝐶 “ (0[,)𝑐)))
64	63	rspceeqv 3605	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ ℝ⁺ ∧ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
65	61, 64	mpan2 701	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ⁺ → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
66	65	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
67	32	metustel 24617	. . . . . . . . . . 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 484	. . . . . . . . 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 259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → (^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
71	19	metustel 24617	. . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → 𝑢 = (^◡𝐶 “ (0[,)𝑐)))
74	73	breqd 5112	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦 ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
75	74	imbi1d 343	. . . . . . . . 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 3227	. . . . . . . 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 5369	. . . . . . 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 3186	. . . . . 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 281	. . . . 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 484	. . . 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 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
82		simplr 778	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑐 ∈ ℝ⁺)
83		simprr 782	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
84		simprl 780	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
85		elbl4 24630	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
86		rpxr 13013	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
87		elbl3ps 24458	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ^*) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
88	86, 87	sylanl2 691	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
89	85, 88	bitr3d 283	. . . . . . . . 9 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
90	81, 82, 83, 84, 89	syl22anc 849	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
91	7	ad4antr 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
92		simpllr 785	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑑 ∈ ℝ⁺)
93		simp-4r 793	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
94	93, 83	ffvelcdmd 7066	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
95	93, 84	ffvelcdmd 7066	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
96		elbl4 24630	. . . . . . . . . 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 13013	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ^*)
98		elbl3ps 24458	. . . . . . . . . . 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 691	. . . . . . . . . 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 283	. . . . . . . . 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 849	. . . . . . . 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 346	. . . . . . 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 3225	. . . . . 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 3185	. . . . 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 3184	. . . 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 281	. . 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 587	. 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 281	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 208 ∧ wa 399 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∀wral 3077 ∃wrex 3087 ∅c0 4286 class class class wbr 5101 ↦ cmpt 5182 × cxp 5646 ^◡ccnv 5647 ran crn 5649 “ cima 5651 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 0cc0 11084 ℝ^*cxr 11226 < clt 11227 ℝ⁺crp 13003 [,)cico 13361 PsMetcpsmet 21415 ballcbl 21418 fBascfbas 21419 filGencfg 21420 metUnifcmetu 21422 UnifOncust 24267 Cn_ucucn 24341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-div 11856 df-nn 12221 df-2 12290 df-rp 13004 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ico 13365 df-psmet 21423 df-bl 21426 df-fbas 21428 df-fg 21429 df-metu 21430 df-fil 23913 df-ust 24268 df-ucn 24342

This theorem is referenced by: qqhucn 34291 heicant 38159

W3C validator