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Theorem metucn 24454
Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24426. (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 24432 . . . . . . 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 2779 . . . . 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 24432 . . . . . . 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 2779 . . . . 5 (𝜑𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))
115, 10oveq12d 7432 . . . 4 (𝜑 → (𝑈 Cnu𝑉) = (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) Cnu((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))))
1211eleq2d 2814 . . 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 2727 . . . 4 ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
14 eqid 2727 . . . 4 ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
15 metucn.x . . . . 5 (𝜑𝑋 ≠ ∅)
16 oveq2 7422 . . . . . . . . 9 (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
1716imaeq2d 6057 . . . . . . . 8 (𝑎 = 𝑐 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑐)))
1817cbvmptv 5255 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
1918rneqi 5933 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
2019metust 24441 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
2115, 2, 20syl2anc 583 . . . 4 (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22 metucn.y . . . . 5 (𝜑𝑌 ≠ ∅)
23 oveq2 7422 . . . . . . . . 9 (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
2423imaeq2d 6057 . . . . . . . 8 (𝑏 = 𝑑 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑑)))
2524cbvmptv 5255 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2625rneqi 5933 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2726metust 24441 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
2822, 7, 27syl2anc 583 . . . 4 (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29 oveq2 7422 . . . . . . . . 9 (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
3029imaeq2d 6057 . . . . . . . 8 (𝑎 = 𝑒 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑒)))
3130cbvmptv 5255 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3231rneqi 5933 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3332metustfbas 24440 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
3415, 2, 33syl2anc 583 . . . 4 (𝜑 → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35 oveq2 7422 . . . . . . . . 9 (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
3635imaeq2d 6057 . . . . . . . 8 (𝑏 = 𝑓 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑓)))
3736cbvmptv 5255 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3837rneqi 5933 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3938metustfbas 24440 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4022, 7, 39syl2anc 583 . . . 4 (𝜑 → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4113, 14, 21, 28, 34, 40isucn2 24158 . . 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 279 . 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 2727 . . . . . . . . . 10 (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))
44 oveq2 7422 . . . . . . . . . . . 12 (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
4544imaeq2d 6057 . . . . . . . . . . 11 (𝑓 = 𝑑 → (𝐷 “ (0[,)𝑓)) = (𝐷 “ (0[,)𝑑)))
4645rspceeqv 3629 . . . . . . . . . 10 ((𝑑 ∈ ℝ+ ∧ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4743, 46mpan2 690 . . . . . . . . 9 (𝑑 ∈ ℝ+ → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4847adantl 481 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4938metustel 24433 . . . . . . . . . 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 480 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
5248, 51mpbird 257 . . . . . . 7 ((𝜑𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
5326metustel 24433 . . . . . . . 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 484 . . . . . . . . . . 11 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → 𝑣 = (𝐷 “ (0[,)𝑑)))
5655breqd 5153 . . . . . . . . . 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 3172 . . . . . . . 8 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
5958rexralbidv 3215 . . . . . . 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 5404 . . . . . 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 2727 . . . . . . . . . . 11 (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))
62 oveq2 7422 . . . . . . . . . . . . 13 (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
6362imaeq2d 6057 . . . . . . . . . . . 12 (𝑒 = 𝑐 → (𝐶 “ (0[,)𝑒)) = (𝐶 “ (0[,)𝑐)))
6463rspceeqv 3629 . . . . . . . . . . 11 ((𝑐 ∈ ℝ+ ∧ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6561, 64mpan2 690 . . . . . . . . . 10 (𝑐 ∈ ℝ+ → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6665adantl 481 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6732metustel 24433 . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
7066, 69mpbird 257 . . . . . . . 8 ((𝜑𝑐 ∈ ℝ+) → (𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
7119metustel 24433 . . . . . . . . 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 484 . . . . . . . . . . 11 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → 𝑢 = (𝐶 “ (0[,)𝑐)))
7473breqd 5153 . . . . . . . . . 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 3213 . . . . . . . 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 5405 . . . . . . 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 3172 . . . . . 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 279 . . . . 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 480 . . . 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 731 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
82 simplr 768 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑐 ∈ ℝ+)
83 simprr 772 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
84 simprl 770 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
85 elbl4 24446 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(𝐶 “ (0[,)𝑐))𝑦))
86 rpxr 13001 . . . . . . . . . . 11 (𝑐 ∈ ℝ+𝑐 ∈ ℝ*)
87 elbl3ps 24271 . . . . . . . . . . 11 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ*) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
8886, 87sylanl2 680 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
8985, 88bitr3d 281 . . . . . . . . 9 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥(𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
9081, 82, 83, 84, 89syl22anc 838 . . . . . . . 8 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
917ad4antr 731 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
92 simpllr 775 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑑 ∈ ℝ+)
93 simp-4r 783 . . . . . . . . . 10 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹:𝑋𝑌)
9493, 83ffvelcdmd 7089 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑦) ∈ 𝑌)
9593, 84ffvelcdmd 7089 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑥) ∈ 𝑌)
96 elbl4 24446 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
97 rpxr 13001 . . . . . . . . . . 11 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
98 elbl3ps 24271 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
9997, 98sylanl2 680 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10096, 99bitr3d 281 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10191, 92, 94, 95, 100syl22anc 838 . . . . . . . 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 3211 . . . . . 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 3171 . . . . 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 3170 . . . 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 279 . . 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 578 . 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 279 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 395   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  c0 4318   class class class wbr 5142  cmpt 5225   × cxp 5670  ccnv 5671  ran crn 5673  cima 5675  wf 6538  cfv 6542  (class class class)co 7414  0cc0 11124  *cxr 11263   < clt 11264  +crp 12992  [,)cico 13344  PsMetcpsmet 21243  ballcbl 21246  fBascfbas 21247  filGencfg 21248  metUnifcmetu 21250  UnifOncust 24078   Cnucucn 24154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-2 12291  df-rp 12993  df-xneg 13110  df-xadd 13111  df-xmul 13112  df-ico 13348  df-psmet 21251  df-bl 21254  df-fbas 21256  df-fg 21257  df-metu 21258  df-fil 23724  df-ust 24079  df-ucn 24155
This theorem is referenced by:  qqhucn  33516  heicant  37050
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