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Theorem metucn 23727

Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 23699. (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 23705	. . . . . . 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 2790	. . . . 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 23705	. . . . . . 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 2790	. . . . 5 ⊢ (𝜑 → 𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))))
11	5, 10	oveq12d 7293	. . . 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 2824	. . 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 2738	. . . 4 ⊢ ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
14		eqid 2738	. . . 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 7283	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
17	16	imaeq2d 5969	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑐)))
18	17	cbvmptv 5187	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
19	18	rneqi 5846	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑐)))
20	19	metust 23714	. . . . 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 7283	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
24	23	imaeq2d 5969	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑑)))
25	24	cbvmptv 5187	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
26	25	rneqi 5846	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
27	26	metust 23714	. . . . 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 7283	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
30	29	imaeq2d 5969	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (^◡𝐶 “ (0[,)𝑎)) = (^◡𝐶 “ (0[,)𝑒)))
31	30	cbvmptv 5187	. . . . . . 7 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
32	31	rneqi 5846	. . . . . 6 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑒)))
33	32	metustfbas 23713	. . . . 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 7283	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
36	35	imaeq2d 5969	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑓)))
37	36	cbvmptv 5187	. . . . . . 7 ⊢ (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
38	37	rneqi 5846	. . . . . 6 ⊢ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑓)))
39	38	metustfbas 23713	. . . . 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 23431	. . 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 278	. 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 2738	. . . . . . . . . 10 ⊢ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))
44		oveq2 7283	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
45	44	imaeq2d 5969	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑑 → (^◡𝐷 “ (0[,)𝑓)) = (^◡𝐷 “ (0[,)𝑑)))
46	45	rspceeqv 3575	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℝ⁺ ∧ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
47	43, 46	mpan2 688	. . . . . . . . 9 ⊢ (𝑑 ∈ ℝ⁺ → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
48	47	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → ∃𝑓 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑑)) = (^◡𝐷 “ (0[,)𝑓)))
49	38	metustel 23706	. . . . . . . . . 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 481	. . . . . . . 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 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏))))
53	26	metustel 23706	. . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → 𝑣 = (^◡𝐷 “ (0[,)𝑑)))
56	55	breqd 5085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦)))
57	56	imbi2d 341	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 = (^◡𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)(^◡𝐷 “ (0[,)𝑑))(𝐹‘𝑦))))
58	57	ralbidv 3112	. . . . . . . 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 3230	. . . . . . 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 5333	. . . . . 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 2738	. . . . . . . . . . 11 ⊢ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))
62		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
63	62	imaeq2d 5969	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑐 → (^◡𝐶 “ (0[,)𝑒)) = (^◡𝐶 “ (0[,)𝑐)))
64	63	rspceeqv 3575	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ ℝ⁺ ∧ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
65	61, 64	mpan2 688	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ⁺ → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
66	65	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → ∃𝑒 ∈ ℝ⁺ (^◡𝐶 “ (0[,)𝑐)) = (^◡𝐶 “ (0[,)𝑒)))
67	32	metustel 23706	. . . . . . . . . . 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 481	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺) → (^◡𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐶 “ (0[,)𝑎))))
71	19	metustel 23706	. . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → 𝑢 = (^◡𝐶 “ (0[,)𝑐)))
74	73	breqd 5085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 = (^◡𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦 ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
75	74	imbi1d 342	. . . . . . . . 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 3129	. . . . . . . 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 5334	. . . . . . 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 3112	. . . . . 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 278	. . . . 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 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))))
81	2	ad4antr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
82		simplr 766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑐 ∈ ℝ⁺)
83		simprr 770	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
84		simprl 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
85		elbl4 23719	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(^◡𝐶 “ (0[,)𝑐))𝑦))
86		rpxr 12739	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
87		elbl3ps 23544	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ^*) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
88	86, 87	sylanl2 678	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
89	85, 88	bitr3d 280	. . . . . . . . 9 ⊢ (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
90	81, 82, 83, 84, 89	syl22anc 836	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(^◡𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
91	7	ad4antr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
92		simpllr 773	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑑 ∈ ℝ⁺)
93		simp-4r 781	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
94	93, 83	ffvelrnd 6962	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
95	93, 84	ffvelrnd 6962	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
96		elbl4 23719	. . . . . . . . . 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 12739	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ^*)
98		elbl3ps 23544	. . . . . . . . . . 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 678	. . . . . . . . . 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 280	. . . . . . . . 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 836	. . . . . . . 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 345	. . . . . . 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 3128	. . . . . 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 3225	. . . . 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 3111	. . . 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 278	. . 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 278	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ∅c0 4256 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ^◡ccnv 5588 ran crn 5590 “ cima 5592 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 0cc0 10871 ℝ^*cxr 11008 < clt 11009 ℝ⁺crp 12730 [,)cico 13081 PsMetcpsmet 20581 ballcbl 20584 fBascfbas 20585 filGencfg 20586 metUnifcmetu 20588 UnifOncust 23351 Cn_ucucn 23427

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-2 12036 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ico 13085 df-psmet 20589 df-bl 20592 df-fbas 20594 df-fg 20595 df-metu 20596 df-fil 22997 df-ust 23352 df-ucn 23428

This theorem is referenced by: qqhucn 31942 heicant 35812

W3C validator