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Theorem metucn 23337
Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 23309. (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 23315 . . . . . . 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, 4syl5eq 2786 . . . . 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 23315 . . . . . . 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, 9syl5eq 2786 . . . . 5 (𝜑𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))
115, 10oveq12d 7201 . . . 4 (𝜑 → (𝑈 Cnu𝑉) = (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) Cnu((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))))
1211eleq2d 2819 . . 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 2739 . . . 4 ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
14 eqid 2739 . . . 4 ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
15 metucn.x . . . . 5 (𝜑𝑋 ≠ ∅)
16 oveq2 7191 . . . . . . . . 9 (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
1716imaeq2d 5913 . . . . . . . 8 (𝑎 = 𝑐 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑐)))
1817cbvmptv 5143 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
1918rneqi 5790 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
2019metust 23324 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
2115, 2, 20syl2anc 587 . . . 4 (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22 metucn.y . . . . 5 (𝜑𝑌 ≠ ∅)
23 oveq2 7191 . . . . . . . . 9 (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
2423imaeq2d 5913 . . . . . . . 8 (𝑏 = 𝑑 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑑)))
2524cbvmptv 5143 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2625rneqi 5790 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2726metust 23324 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
2822, 7, 27syl2anc 587 . . . 4 (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29 oveq2 7191 . . . . . . . . 9 (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
3029imaeq2d 5913 . . . . . . . 8 (𝑎 = 𝑒 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑒)))
3130cbvmptv 5143 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3231rneqi 5790 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3332metustfbas 23323 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
3415, 2, 33syl2anc 587 . . . 4 (𝜑 → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35 oveq2 7191 . . . . . . . . 9 (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
3635imaeq2d 5913 . . . . . . . 8 (𝑏 = 𝑓 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑓)))
3736cbvmptv 5143 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3837rneqi 5790 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3938metustfbas 23323 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4022, 7, 39syl2anc 587 . . . 4 (𝜑 → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4113, 14, 21, 28, 34, 40isucn2 23044 . . 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 282 . 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 2739 . . . . . . . . . 10 (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))
44 oveq2 7191 . . . . . . . . . . . 12 (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
4544imaeq2d 5913 . . . . . . . . . . 11 (𝑓 = 𝑑 → (𝐷 “ (0[,)𝑓)) = (𝐷 “ (0[,)𝑑)))
4645rspceeqv 3544 . . . . . . . . . 10 ((𝑑 ∈ ℝ+ ∧ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4743, 46mpan2 691 . . . . . . . . 9 (𝑑 ∈ ℝ+ → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4847adantl 485 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4938metustel 23316 . . . . . . . . . 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 484 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
5248, 51mpbird 260 . . . . . . 7 ((𝜑𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
5326metustel 23316 . . . . . . . 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 488 . . . . . . . . . . 11 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → 𝑣 = (𝐷 “ (0[,)𝑑)))
5655breqd 5051 . . . . . . . . . 10 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → ((𝐹𝑥)𝑣(𝐹𝑦) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
5756imbi2d 344 . . . . . . . . 9 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
5857ralbidv 3110 . . . . . . . 8 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
5958rexralbidv 3212 . . . . . . 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 5287 . . . . . 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 2739 . . . . . . . . . . 11 (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))
62 oveq2 7191 . . . . . . . . . . . . 13 (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
6362imaeq2d 5913 . . . . . . . . . . . 12 (𝑒 = 𝑐 → (𝐶 “ (0[,)𝑒)) = (𝐶 “ (0[,)𝑐)))
6463rspceeqv 3544 . . . . . . . . . . 11 ((𝑐 ∈ ℝ+ ∧ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6561, 64mpan2 691 . . . . . . . . . 10 (𝑐 ∈ ℝ+ → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6665adantl 485 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6732metustel 23316 . . . . . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
7066, 69mpbird 260 . . . . . . . 8 ((𝜑𝑐 ∈ ℝ+) → (𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
7119metustel 23316 . . . . . . . . 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 488 . . . . . . . . . . 11 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → 𝑢 = (𝐶 “ (0[,)𝑐)))
7473breqd 5051 . . . . . . . . . 10 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦𝑥(𝐶 “ (0[,)𝑐))𝑦))
7574imbi1d 345 . . . . . . . . 9 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → ((𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
76752ralbidv 3112 . . . . . . . 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 5288 . . . . . . 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 3110 . . . . . 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 282 . . . . 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 484 . . . 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 732 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
82 simplr 769 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑐 ∈ ℝ+)
83 simprr 773 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
84 simprl 771 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
85 elbl4 23329 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(𝐶 “ (0[,)𝑐))𝑦))
86 rpxr 12494 . . . . . . . . . . 11 (𝑐 ∈ ℝ+𝑐 ∈ ℝ*)
87 elbl3ps 23157 . . . . . . . . . . 11 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ*) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
8886, 87sylanl2 681 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
8985, 88bitr3d 284 . . . . . . . . 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 732 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
92 simpllr 776 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑑 ∈ ℝ+)
93 simp-4r 784 . . . . . . . . . 10 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹:𝑋𝑌)
9493, 83ffvelrnd 6875 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑦) ∈ 𝑌)
9593, 84ffvelrnd 6875 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑥) ∈ 𝑌)
96 elbl4 23329 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
97 rpxr 12494 . . . . . . . . . . 11 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
98 elbl3ps 23157 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
9997, 98sylanl2 681 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10096, 99bitr3d 284 . . . . . . . . 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 348 . . . . . . 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 3111 . . . . . 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 3207 . . . . 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 3109 . . . 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 282 . . 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 582 . 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 282 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 209  wa 399   = wceq 1542  wcel 2114  wne 2935  wral 3054  wrex 3055  c0 4221   class class class wbr 5040  cmpt 5120   × cxp 5533  ccnv 5534  ran crn 5536  cima 5538  wf 6346  cfv 6350  (class class class)co 7183  0cc0 10628  *cxr 10765   < clt 10766  +crp 12485  [,)cico 12836  PsMetcpsmet 20214  ballcbl 20217  fBascfbas 20218  filGencfg 20219  metUnifcmetu 20221  UnifOncust 22964   Cnucucn 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-cnex 10684  ax-resscn 10685  ax-1cn 10686  ax-icn 10687  ax-addcl 10688  ax-addrcl 10689  ax-mulcl 10690  ax-mulrcl 10691  ax-mulcom 10692  ax-addass 10693  ax-mulass 10694  ax-distr 10695  ax-i2m1 10696  ax-1ne0 10697  ax-1rid 10698  ax-rnegex 10699  ax-rrecex 10700  ax-cnre 10701  ax-pre-lttri 10702  ax-pre-lttrn 10703  ax-pre-ltadd 10704  ax-pre-mulgt0 10705
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-po 5452  df-so 5453  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7140  df-ov 7186  df-oprab 7187  df-mpo 7188  df-1st 7727  df-2nd 7728  df-er 8333  df-map 8452  df-en 8569  df-dom 8570  df-sdom 8571  df-pnf 10768  df-mnf 10769  df-xr 10770  df-ltxr 10771  df-le 10772  df-sub 10963  df-neg 10964  df-div 11389  df-2 11792  df-rp 12486  df-xneg 12603  df-xadd 12604  df-xmul 12605  df-ico 12840  df-psmet 20222  df-bl 20225  df-fbas 20227  df-fg 20228  df-metu 20229  df-fil 22610  df-ust 22965  df-ucn 23041
This theorem is referenced by:  qqhucn  31525  heicant  35468
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