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Theorem ucnima 22409
Description: An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1 (𝜑𝑈 ∈ (UnifOn‘𝑋))
ucnprima.2 (𝜑𝑉 ∈ (UnifOn‘𝑌))
ucnprima.3 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
ucnprima.4 (𝜑𝑊𝑉)
ucnprima.5 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
Assertion
Ref Expression
ucnima (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑋,𝑦,𝑟   𝐹,𝑟   𝑥,𝐺,𝑦   𝑈,𝑟,𝑥,𝑦   𝑉,𝑟,𝑥   𝑊,𝑟,𝑥,𝑦   𝑋,𝑟   𝑌,𝑟,𝑥   𝜑,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐺(𝑟)   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem ucnima
Dummy variables 𝑝 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4843 . . . . . . . 8 (𝑤 = 𝑊 → ((𝐹𝑥)𝑤(𝐹𝑦) ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
21imbi2d 332 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
32ralbidv 3165 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
43rexralbidv 3237 . . . . 5 (𝑤 = 𝑊 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
5 ucnprima.3 . . . . . . 7 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
6 ucnprima.1 . . . . . . . 8 (𝜑𝑈 ∈ (UnifOn‘𝑋))
7 ucnprima.2 . . . . . . . 8 (𝜑𝑉 ∈ (UnifOn‘𝑌))
8 isucn 22406 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))))
96, 7, 8syl2anc 580 . . . . . . 7 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))))
105, 9mpbid 224 . . . . . 6 (𝜑 → (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦))))
1110simprd 490 . . . . 5 (𝜑 → ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))
12 ucnprima.4 . . . . 5 (𝜑𝑊𝑉)
134, 11, 12rspcdva 3501 . . . 4 (𝜑 → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
14 simplll 792 . . . . . . . . 9 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝜑)
15 simplr 786 . . . . . . . . 9 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
1614, 15jca 508 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → (𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
17 ustssxp 22332 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
186, 17sylan 576 . . . . . . . . . 10 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
1918sselda 3796 . . . . . . . . 9 (((𝜑𝑟𝑈) ∧ 𝑝𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
2019adantlr 707 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
21 simpr 478 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝑝𝑟)
22 simplr 786 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
23 simpr 478 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → 𝑝 ∈ (𝑋 × 𝑋))
24 elxp2 5334 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝑋 × 𝑋) ↔ ∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
2523, 24sylib 210 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
26 simpr 478 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
2726eleq1d 2861 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
2827adantlr 707 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
29 df-br 4842 . . . . . . . . . . . . . . . . . 18 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
3028, 29syl6bbr 281 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟𝑥𝑟𝑦))
31 simplr 786 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 ∈ (𝑋 × 𝑋))
32 opex 5121 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V
33 ucnprima.5 . . . . . . . . . . . . . . . . . . . . . . 23 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
346, 7, 5, 12, 33ucnimalem 22408 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3534fvmpt2 6514 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ (𝑋 × 𝑋) ∧ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V) → (𝐺𝑝) = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3631, 32, 35sylancl 581 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
37 simpr 478 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
38 1st2nd2 7438 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 ∈ (𝑋 × 𝑋) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
3931, 38syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
4037, 39eqtr3d 2833 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
41 vex 3386 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑥 ∈ V
42 vex 3386 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 ∈ V
4341, 42opth 5133 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩ ↔ (𝑥 = (1st𝑝) ∧ 𝑦 = (2nd𝑝)))
4440, 43sylib 210 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑥 = (1st𝑝) ∧ 𝑦 = (2nd𝑝)))
4544simpld 489 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑥 = (1st𝑝))
4645fveq2d 6413 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹𝑥) = (𝐹‘(1st𝑝)))
4744simprd 490 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑦 = (2nd𝑝))
4847fveq2d 6413 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹𝑦) = (𝐹‘(2nd𝑝)))
4946, 48opeq12d 4599 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
5036, 49eqtr4d 2834 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
5150eleq1d 2861 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊))
52 df-br 4842 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥)𝑊(𝐹𝑦) ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊)
5351, 52syl6bbr 281 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
5430, 53imbi12d 336 . . . . . . . . . . . . . . . 16 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝑝𝑟 → (𝐺𝑝) ∈ 𝑊) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
5554exbiri 846 . . . . . . . . . . . . . . 15 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5655reximdv 3194 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5756reximdv 3194 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5825, 57mpd 15 . . . . . . . . . . . 12 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊)))
5958adantlr 707 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊)))
6022, 59r19.29d2r 3259 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
61 pm3.35 838 . . . . . . . . . . . 12 (((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6261rexlimivw 3208 . . . . . . . . . . 11 (∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6362rexlimivw 3208 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6460, 63syl 17 . . . . . . . . 9 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6564imp 396 . . . . . . . 8 ((((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6616, 20, 21, 65syl21anc 867 . . . . . . 7 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6766ralrimiva 3145 . . . . . 6 (((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
6867ex 402 . . . . 5 ((𝜑𝑟𝑈) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
6968reximdva 3195 . . . 4 (𝜑 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7013, 69mpd 15 . . 3 (𝜑 → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
7133mpt2fun 6994 . . . . . 6 Fun 𝐺
72 opex 5121 . . . . . . . 8 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
7333, 72dmmpt2 7474 . . . . . . 7 dom 𝐺 = (𝑋 × 𝑋)
7418, 73syl6sseqr 3846 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
75 funimass4 6470 . . . . . 6 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊 ↔ ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7671, 74, 75sylancr 582 . . . . 5 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊 ↔ ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7776biimprd 240 . . . 4 ((𝜑𝑟𝑈) → (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
7877ralrimiva 3145 . . 3 (𝜑 → ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
79 r19.29r 3252 . . 3 ((∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → ∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)))
8070, 78, 79syl2anc 580 . 2 (𝜑 → ∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)))
81 pm3.35 838 . . 3 ((∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → (𝐺𝑟) ⊆ 𝑊)
8281reximi 3189 . 2 (∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
8380, 82syl 17 1 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3087  wrex 3088  Vcvv 3383  wss 3767  cop 4372   class class class wbr 4841   × cxp 5308  dom cdm 5310  cima 5313  Fun wfun 6093  wf 6095  cfv 6099  (class class class)co 6876  cmpt2 6878  1st c1st 7397  2nd c2nd 7398  UnifOncust 22327   Cnucucn 22403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-map 8095  df-ust 22328  df-ucn 22404
This theorem is referenced by:  ucnprima  22410
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