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Theorem ucnima 24305
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 5149 . . . . . . . 8 (𝑤 = 𝑊 → ((𝐹𝑥)𝑤(𝐹𝑦) ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
21imbi2d 340 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
32ralbidv 3175 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
43rexralbidv 3220 . . . . 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 24302 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))))
96, 7, 8syl2anc 584 . . . . . . 7 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))))
105, 9mpbid 232 . . . . . 6 (𝜑 → (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦))))
1110simprd 495 . . . . 5 (𝜑 → ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))
12 ucnprima.4 . . . . 5 (𝜑𝑊𝑉)
134, 11, 12rspcdva 3622 . . . 4 (𝜑 → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
14 simplll 775 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝜑)
15 simplr 769 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
16 ustssxp 24228 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
176, 16sylan 580 . . . . . . . . . 10 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
1817sselda 3994 . . . . . . . . 9 (((𝜑𝑟𝑈) ∧ 𝑝𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
1918adantlr 715 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
20 simpr 484 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝑝𝑟)
21 simplr 769 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
22 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → 𝑝 ∈ (𝑋 × 𝑋))
23 elxp2 5712 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝑋 × 𝑋) ↔ ∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
2422, 23sylib 218 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
25 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
2625eleq1d 2823 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
2726adantlr 715 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
28 df-br 5148 . . . . . . . . . . . . . . . . . 18 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
2927, 28bitr4di 289 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟𝑥𝑟𝑦))
30 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 ∈ (𝑋 × 𝑋))
31 opex 5474 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V
32 ucnprima.5 . . . . . . . . . . . . . . . . . . . . . . 23 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
336, 7, 5, 12, 32ucnimalem 24304 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3433fvmpt2 7026 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ (𝑋 × 𝑋) ∧ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V) → (𝐺𝑝) = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3530, 31, 34sylancl 586 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
36 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
37 1st2nd2 8051 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 ∈ (𝑋 × 𝑋) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
3830, 37syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
3936, 38eqtr3d 2776 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
40 vex 3481 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑥 ∈ V
41 vex 3481 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 ∈ V
4240, 41opth 5486 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩ ↔ (𝑥 = (1st𝑝) ∧ 𝑦 = (2nd𝑝)))
4339, 42sylib 218 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑥 = (1st𝑝) ∧ 𝑦 = (2nd𝑝)))
4443simpld 494 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑥 = (1st𝑝))
4544fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹𝑥) = (𝐹‘(1st𝑝)))
4643simprd 495 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑦 = (2nd𝑝))
4746fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹𝑦) = (𝐹‘(2nd𝑝)))
4845, 47opeq12d 4885 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
4935, 48eqtr4d 2777 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
5049eleq1d 2823 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊))
51 df-br 5148 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥)𝑊(𝐹𝑦) ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊)
5250, 51bitr4di 289 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
5329, 52imbi12d 344 . . . . . . . . . . . . . . . 16 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝑝𝑟 → (𝐺𝑝) ∈ 𝑊) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
5453exbiri 811 . . . . . . . . . . . . . . 15 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5554reximdv 3167 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5655reximdv 3167 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5724, 56mpd 15 . . . . . . . . . . . 12 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊)))
5857adantlr 715 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊)))
5921, 58r19.29d2r 3137 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
60 pm3.35 803 . . . . . . . . . . . 12 (((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6160rexlimivw 3148 . . . . . . . . . . 11 (∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6261rexlimivw 3148 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6359, 62syl 17 . . . . . . . . 9 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6463imp 406 . . . . . . . 8 ((((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6514, 15, 19, 20, 64syl1111anc 840 . . . . . . 7 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6665ralrimiva 3143 . . . . . 6 (((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
6766ex 412 . . . . 5 ((𝜑𝑟𝑈) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
6867reximdva 3165 . . . 4 (𝜑 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
6913, 68mpd 15 . . 3 (𝜑 → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
7032mpofun 7556 . . . . . 6 Fun 𝐺
71 opex 5474 . . . . . . . 8 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
7232, 71dmmpo 8094 . . . . . . 7 dom 𝐺 = (𝑋 × 𝑋)
7317, 72sseqtrrdi 4046 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
74 funimass4 6972 . . . . . 6 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊 ↔ ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7570, 73, 74sylancr 587 . . . . 5 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊 ↔ ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7675biimprd 248 . . . 4 ((𝜑𝑟𝑈) → (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
7776ralrimiva 3143 . . 3 (𝜑 → ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
78 r19.29r 3113 . . 3 ((∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → ∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)))
7969, 77, 78syl2anc 584 . 2 (𝜑 → ∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)))
80 pm3.35 803 . . 3 ((∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → (𝐺𝑟) ⊆ 𝑊)
8180reximi 3081 . 2 (∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
8279, 81syl 17 1 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  wss 3962  cop 4636   class class class wbr 5147   × cxp 5686  dom cdm 5688  cima 5691  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  UnifOncust 24223   Cnucucn 24299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-ust 24224  df-ucn 24300
This theorem is referenced by:  ucnprima  24306
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