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Theorem metucn 22186
 Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 22158. (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 22164 . . . . . . 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 2656 . . . . 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 22164 . . . . . . 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 2656 . . . . 5 (𝜑𝑉 = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))))
115, 10oveq12d 6567 . . . 4 (𝜑 → (𝑈 Cnu𝑉) = (((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) Cnu((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))))
1211eleq2d 2673 . . 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 2610 . . . 4 ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
14 eqid 2610 . . . 4 ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) = ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
15 metucn.x . . . . 5 (𝜑𝑋 ≠ ∅)
16 oveq2 6557 . . . . . . . . 9 (𝑎 = 𝑐 → (0[,)𝑎) = (0[,)𝑐))
1716imaeq2d 5385 . . . . . . . 8 (𝑎 = 𝑐 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑐)))
1817cbvmptv 4678 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
1918rneqi 5273 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑐 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑐)))
2019metust 22173 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
2115, 2, 20syl2anc 691 . . . 4 (𝜑 → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎)))) ∈ (UnifOn‘𝑋))
22 metucn.y . . . . 5 (𝜑𝑌 ≠ ∅)
23 oveq2 6557 . . . . . . . . 9 (𝑏 = 𝑑 → (0[,)𝑏) = (0[,)𝑑))
2423imaeq2d 5385 . . . . . . . 8 (𝑏 = 𝑑 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑑)))
2524cbvmptv 4678 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2625rneqi 5273 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
2726metust 22173 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
2822, 7, 27syl2anc 691 . . . 4 (𝜑 → ((𝑌 × 𝑌)filGenran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))) ∈ (UnifOn‘𝑌))
29 oveq2 6557 . . . . . . . . 9 (𝑎 = 𝑒 → (0[,)𝑎) = (0[,)𝑒))
3029imaeq2d 5385 . . . . . . . 8 (𝑎 = 𝑒 → (𝐶 “ (0[,)𝑎)) = (𝐶 “ (0[,)𝑒)))
3130cbvmptv 4678 . . . . . . 7 (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3231rneqi 5273 . . . . . 6 ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) = ran (𝑒 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑒)))
3332metustfbas 22172 . . . . 5 ((𝑋 ≠ ∅ ∧ 𝐶 ∈ (PsMet‘𝑋)) → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
3415, 2, 33syl2anc 691 . . . 4 (𝜑 → ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ∈ (fBas‘(𝑋 × 𝑋)))
35 oveq2 6557 . . . . . . . . 9 (𝑏 = 𝑓 → (0[,)𝑏) = (0[,)𝑓))
3635imaeq2d 5385 . . . . . . . 8 (𝑏 = 𝑓 → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑓)))
3736cbvmptv 4678 . . . . . . 7 (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3837rneqi 5273 . . . . . 6 ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) = ran (𝑓 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑓)))
3938metustfbas 22172 . . . . 5 ((𝑌 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑌)) → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4022, 7, 39syl2anc 691 . . . 4 (𝜑 → ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ∈ (fBas‘(𝑌 × 𝑌)))
4113, 14, 21, 28, 34, 40isucn2 21893 . . 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 267 . 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 2610 . . . . . . . . . 10 (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))
44 oveq2 6557 . . . . . . . . . . . . 13 (𝑓 = 𝑑 → (0[,)𝑓) = (0[,)𝑑))
4544imaeq2d 5385 . . . . . . . . . . . 12 (𝑓 = 𝑑 → (𝐷 “ (0[,)𝑓)) = (𝐷 “ (0[,)𝑑)))
4645eqeq2d 2620 . . . . . . . . . . 11 (𝑓 = 𝑑 → ((𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)) ↔ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))))
4746rspcev 3282 . . . . . . . . . 10 ((𝑑 ∈ ℝ+ ∧ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑑))) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4843, 47mpan2 703 . . . . . . . . 9 (𝑑 ∈ ℝ+ → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
4948adantl 481 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓)))
5038metustel 22165 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑌) → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
517, 50syl 17 . . . . . . . . 9 (𝜑 → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
5251adantr 480 . . . . . . . 8 ((𝜑𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑓 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑓))))
5349, 52mpbird 246 . . . . . . 7 ((𝜑𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))))
5426metustel 22165 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑌) → (𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ+ 𝑣 = (𝐷 “ (0[,)𝑑))))
557, 54syl 17 . . . . . . 7 (𝜑 → (𝑣 ∈ ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏))) ↔ ∃𝑑 ∈ ℝ+ 𝑣 = (𝐷 “ (0[,)𝑑))))
56 simpr 476 . . . . . . . . . . 11 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → 𝑣 = (𝐷 “ (0[,)𝑑)))
5756breqd 4594 . . . . . . . . . 10 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → ((𝐹𝑥)𝑣(𝐹𝑦) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
5857imbi2d 329 . . . . . . . . 9 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
5958ralbidv 2969 . . . . . . . 8 ((𝜑𝑣 = (𝐷 “ (0[,)𝑑))) → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
6059rexralbidv 3040 . . . . . . 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))))
6153, 55, 60ralxfr2d 4808 . . . . . 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))))
62 eqid 2610 . . . . . . . . . . 11 (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))
63 oveq2 6557 . . . . . . . . . . . . . 14 (𝑒 = 𝑐 → (0[,)𝑒) = (0[,)𝑐))
6463imaeq2d 5385 . . . . . . . . . . . . 13 (𝑒 = 𝑐 → (𝐶 “ (0[,)𝑒)) = (𝐶 “ (0[,)𝑐)))
6564eqeq2d 2620 . . . . . . . . . . . 12 (𝑒 = 𝑐 → ((𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)) ↔ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))))
6665rspcev 3282 . . . . . . . . . . 11 ((𝑐 ∈ ℝ+ ∧ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑐))) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6762, 66mpan2 703 . . . . . . . . . 10 (𝑐 ∈ ℝ+ → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6867adantl 481 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒)))
6932metustel 22165 . . . . . . . . . . 11 (𝐶 ∈ (PsMet‘𝑋) → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
702, 69syl 17 . . . . . . . . . 10 (𝜑 → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
7170adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ ℝ+) → ((𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑒 ∈ ℝ+ (𝐶 “ (0[,)𝑐)) = (𝐶 “ (0[,)𝑒))))
7268, 71mpbird 246 . . . . . . . 8 ((𝜑𝑐 ∈ ℝ+) → (𝐶 “ (0[,)𝑐)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))))
7319metustel 22165 . . . . . . . . 9 (𝐶 ∈ (PsMet‘𝑋) → (𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ+ 𝑢 = (𝐶 “ (0[,)𝑐))))
742, 73syl 17 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐶 “ (0[,)𝑎))) ↔ ∃𝑐 ∈ ℝ+ 𝑢 = (𝐶 “ (0[,)𝑐))))
75 simpr 476 . . . . . . . . . . 11 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → 𝑢 = (𝐶 “ (0[,)𝑐)))
7675breqd 4594 . . . . . . . . . 10 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → (𝑥𝑢𝑦𝑥(𝐶 “ (0[,)𝑐))𝑦))
7776imbi1d 330 . . . . . . . . 9 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → ((𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
78772ralbidv 2972 . . . . . . . 8 ((𝜑𝑢 = (𝐶 “ (0[,)𝑐))) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦))))
7972, 74, 78rexxfr2d 4809 . . . . . . 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))))
8079ralbidv 2969 . . . . . 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))))
8161, 80bitrd 267 . . . . 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))))
8281adantr 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))))
832ad4antr 764 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐶 ∈ (PsMet‘𝑋))
84 simplr 788 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑐 ∈ ℝ+)
85 simprr 792 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
86 simprl 790 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
87 elbl4 22178 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ 𝑥(𝐶 “ (0[,)𝑐))𝑦))
88 rpxr 11716 . . . . . . . . . . 11 (𝑐 ∈ ℝ+𝑐 ∈ ℝ*)
89 elbl3ps 22006 . . . . . . . . . . 11 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ*) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
9088, 89sylanl2 681 . . . . . . . . . 10 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥 ∈ (𝑦(ball‘𝐶)𝑐) ↔ (𝑥𝐶𝑦) < 𝑐))
9187, 90bitr3d 269 . . . . . . . . 9 (((𝐶 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑦𝑋𝑥𝑋)) → (𝑥(𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
9283, 84, 85, 86, 91syl22anc 1319 . . . . . . . 8 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(𝐶 “ (0[,)𝑐))𝑦 ↔ (𝑥𝐶𝑦) < 𝑐))
937ad4antr 764 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐷 ∈ (PsMet‘𝑌))
94 simpllr 795 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝑑 ∈ ℝ+)
95 simp-4r 803 . . . . . . . . . 10 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹:𝑋𝑌)
9695, 85ffvelrnd 6268 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑦) ∈ 𝑌)
9795, 86ffvelrnd 6268 . . . . . . . . 9 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑥) ∈ 𝑌)
98 elbl4 22178 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)))
99 rpxr 11716 . . . . . . . . . . 11 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
100 elbl3ps 22006 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ*) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10199, 100sylanl2 681 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥) ∈ ((𝐹𝑦)(ball‘𝐷)𝑑) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10298, 101bitr3d 269 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ((𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑥) ∈ 𝑌)) → ((𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10393, 94, 96, 97, 102syl22anc 1319 . . . . . . . 8 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦) ↔ ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))
10492, 103imbi12d 333 . . . . . . 7 (((((𝜑𝐹:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑐 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥(𝐶 “ (0[,)𝑐))𝑦 → (𝐹𝑥)(𝐷 “ (0[,)𝑑))(𝐹𝑦)) ↔ ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑)))
1051042ralbidva 2971 . . . . . 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)))
106105rexbidva 3031 . . . . 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)))
107106ralbidva 2968 . . . 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)))
10882, 107bitrd 267 . . 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)))
109108pm5.32da 671 . 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))))
11042, 109bitrd 267 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ran crn 5039   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℝ*cxr 9952   < clt 9953  ℝ+crp 11708  [,)cico 12048  PsMetcpsmet 19551  ballcbl 19554  fBascfbas 19555  filGencfg 19556  metUnifcmetu 19558  UnifOncust 21813   Cnucucn 21889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-2 10956  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-psmet 19559  df-bl 19562  df-fbas 19564  df-fg 19565  df-metu 19566  df-fil 21460  df-ust 21814  df-ucn 21890 This theorem is referenced by:  qqhucn  29364  heicant  32614
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