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Theorem metucn 23927
Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 23899. (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 (𝜑𝑋 ≠ ∅)
metucn.y (𝜑𝑌 ≠ ∅)
metucn.c (𝜑𝐶 ∈ (PsMet‘𝑋))
metucn.d (𝜑𝐷 ∈ (PsMet‘𝑌))
Assertion
Ref Expression
metucn (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 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.
StepHypRef Expression
1 metucn.u . . . . . 6 𝑈 = (metUnif‘𝐶)
2 metucn.c . . . . . . 7 (𝜑𝐶 ∈ (PsMet‘𝑋))
3 metuval 23905 . . . . . . 7 (𝐶 ∈ (PsMet‘𝑋) → (metUnif‘𝐶) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))))
42, 3syl 17 . . . . . 6 (𝜑 → (metUnif‘𝐶) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))))
51, 4eqtrid 2788 . . . . 5 (𝜑𝑈 = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))))
6 metucn.v . . . . . 6 𝑉 = (metUnif‘𝐷)
7 metucn.d . . . . . . 7 (𝜑𝐷 ∈ (PsMet‘𝑌))
8 metuval 23905 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑌) → (metUnif‘𝐷) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))
97, 8syl 17 . . . . . 6 (𝜑 → (metUnif‘𝐷) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))
106, 9eqtrid 2788 . . . . 5 (𝜑𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))
115, 10oveq12d 7375 . . . 4 (𝜑 → (𝑈 Cnu𝑉) = (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) Cnu((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))))
1211eleq2d 2823 . . 3 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ 𝐹 ∈ (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) Cnu((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))))
13 eqid 2736 . . . 4 ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
14 eqid 2736 . . . 4 ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
15 metucn.x . . . . 5 (𝜑𝑋 ≠ ∅)
16 oveq2 7365 . . . . . . . . 9 (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
1716imaeq2d 6013 . . . . . . . 8 (𝑎 = 𝑐 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑐)))
1817cbvmptv 5218 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
1918rneqi 5892 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
2019metust 23914 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
2115, 2, 20syl2anc 584 . . . 4 (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22 metucn.y . . . . 5 (𝜑𝑌 ≠ ∅)
23 oveq2 7365 . . . . . . . . 9 (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
2423imaeq2d 6013 . . . . . . . 8 (𝑏 = 𝑑 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑑)))
2524cbvmptv 5218 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2625rneqi 5892 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2726metust 23914 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
2822, 7, 27syl2anc 584 . . . 4 (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29 oveq2 7365 . . . . . . . . 9 (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
3029imaeq2d 6013 . . . . . . . 8 (𝑎 = 𝑒 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑒)))
3130cbvmptv 5218 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3231rneqi 5892 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3332metustfbas 23913 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
3415, 2, 33syl2anc 584 . . . 4 (𝜑 → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35 oveq2 7365 . . . . . . . . 9 (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
3635imaeq2d 6013 . . . . . . . 8 (𝑏 = 𝑓 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑓)))
3736cbvmptv 5218 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3837rneqi 5892 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3938metustfbas 23913 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4022, 7, 39syl2anc 584 . . . 4 (𝜑 → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4113, 14, 21, 28, 34, 40isucn2 23631 . . 3 (𝜑 → (𝐹 ∈ (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) Cnu((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
4212, 41bitrd 278 . 2 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
43 eqid 2736 . . . . . . . . . 10 (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))
44 oveq2 7365 . . . . . . . . . . . 12 (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
4544imaeq2d 6013 . . . . . . . . . . 11 (𝑓 = 𝑑 → (𝐷 “ (0[,)𝑓)) = (𝐷 “ (0[,)𝑑)))
4645rspceeqv 3595 . . . . . . . . . 10 ((𝑑 ∈ ℝ+ ∧ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4743, 46mpan2 689 . . . . . . . . 9 (𝑑 ∈ ℝ+ → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4847adantl 482 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4938metustel 23906 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑌) → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
507, 49syl 17 . . . . . . . . 9 (𝜑 → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
5150adantr 481 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
5248, 51mpbird 256 . . . . . . 7 ((𝜑𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
5326metustel 23906 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑌) → (𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ+ 𝑣 = (𝐷 “ (0[,)𝑑))))
547, 53syl 17 . . . . . . 7 (𝜑 → (𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ+ 𝑣 = (𝐷 “ (0[,)𝑑))))
55 simpr 485 . . . . . . . . . . 11 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → 𝑣 = (𝐷 “ (0[,)𝑑)))
5655breqd 5116 . . . . . . . . . 10 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → ((𝐹𝑥)𝑣(𝐹𝑦) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
5756imbi2d 340 . . . . . . . . 9 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
5857ralbidv 3174 . . . . . . . 8 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
5958rexralbidv 3214 . . . . . . 7 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → (∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
6052, 54, 59ralxfr2d 5365 . . . . . 6 (𝜑 → (∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑑 ∈ ℝ+𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
61 eqid 2736 . . . . . . . . . . 11 (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))
62 oveq2 7365 . . . . . . . . . . . . 13 (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
6362imaeq2d 6013 . . . . . . . . . . . 12 (𝑒 = 𝑐 → (𝐶 “ (0[,)𝑒)) = (𝐶 “ (0[,)𝑐)))
6463rspceeqv 3595 . . . . . . . . . . 11 ((𝑐 ∈ ℝ+ ∧ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6561, 64mpan2 689 . . . . . . . . . 10 (𝑐 ∈ ℝ+ → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6665adantl 482 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6732metustel 23906 . . . . . . . . . . 11 (𝐶 ∈ (PsMet‘𝑋) → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
682, 67syl 17 . . . . . . . . . 10 (𝜑 → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
6968adantr 481 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
7066, 69mpbird 256 . . . . . . . 8 ((𝜑𝑐 ∈ ℝ+) → (𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
7119metustel 23906 . . . . . . . . 9 (𝐶 ∈ (PsMet‘𝑋) → (𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ+ 𝑢 = (𝐶 “ (0[,)𝑐))))
722, 71syl 17 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ+ 𝑢 = (𝐶 “ (0[,)𝑐))))
73 simpr 485 . . . . . . . . . . 11 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → 𝑢 = (𝐶 “ (0[,)𝑐)))
7473breqd 5116 . . . . . . . . . 10 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦𝑥(𝐶 “ (0[,)𝑐))𝑦))
7574imbi1d 341 . . . . . . . . 9 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → ((𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
76752ralbidv 3212 . . . . . . . 8 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
7770, 72, 76rexxfr2d 5366 . . . . . . 7 (𝜑 → (∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∃𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
7877ralbidv 3174 . . . . . 6 (𝜑 → (∀𝑑 ∈ ℝ+𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
7960, 78bitrd 278 . . . . 5 (𝜑 → (∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
8079adantr 481 . . . 4 ((𝜑𝐹:𝑋𝑌) → (∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
812ad4antr 730 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
82 simplr 767 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑐 ∈ ℝ+)
83 simprr 771 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
84 simprl 769 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
85 elbl4 23919 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(𝐶 “ (0[,)𝑐))𝑦))
86 rpxr 12924 . . . . . . . . . . 11 (𝑐 ∈ ℝ+𝑐 ∈ ℝ*)
87 elbl3ps 23744 . . . . . . . . . . 11 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ*) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
8886, 87sylanl2 679 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
8985, 88bitr3d 280 . . . . . . . . 9 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥(𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
9081, 82, 83, 84, 89syl22anc 837 . . . . . . . 8 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
917ad4antr 730 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
92 simpllr 774 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑑 ∈ ℝ+)
93 simp-4r 782 . . . . . . . . . 10 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹:𝑋𝑌)
9493, 83ffvelcdmd 7036 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑦) ∈ 𝑌)
9593, 84ffvelcdmd 7036 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑥) ∈ 𝑌)
96 elbl4 23919 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
97 rpxr 12924 . . . . . . . . . . 11 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
98 elbl3ps 23744 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
9997, 98sylanl2 679 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10096, 99bitr3d 280 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10191, 92, 94, 95, 100syl22anc 837 . . . . . . . 8 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10290, 101imbi12d 344 . . . . . . 7 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑)))
1031022ralbidva 3210 . . . . . 6 ((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) → (∀𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑)))
104103rexbidva 3173 . . . . 5 (((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → (∃𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∃𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑)))
105104ralbidva 3172 . . . 4 ((𝜑𝐹:𝑋𝑌) → (∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑)))
10680, 105bitrd 278 . . 3 ((𝜑𝐹:𝑋𝑌) → (∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑)))
107106pm5.32da 579 . 2 (𝜑 → ((𝐹:𝑋𝑌 ∧ ∀𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))∃𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))))
10842, 107bitrd 278 1 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  c0 4282   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  *cxr 11188   < clt 11189  +crp 12915  [,)cico 13266  PsMetcpsmet 20780  ballcbl 20783  fBascfbas 20784  filGencfg 20785  metUnifcmetu 20787  UnifOncust 23551   Cnucucn 23627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-2 12216  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-psmet 20788  df-bl 20791  df-fbas 20793  df-fg 20794  df-metu 20795  df-fil 23197  df-ust 23552  df-ucn 23628
This theorem is referenced by:  qqhucn  32573  heicant  36113
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