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Theorem ucnima 24290
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 5145 . . . . . . . 8 (𝑤 = 𝑊 → ((𝐹𝑥)𝑤(𝐹𝑦) ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
21imbi2d 340 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
32ralbidv 3178 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
43rexralbidv 3223 . . . . 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 24287 . . . . . . . 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 3623 . . . 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 24213 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
176, 16sylan 580 . . . . . . . . . 10 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
1817sselda 3983 . . . . . . . . 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 5709 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
2726adantlr 715 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
28 df-br 5144 . . . . . . . . . . . . . . . . . 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 5469 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V
32 ucnprima.5 . . . . . . . . . . . . . . . . . . . . . . 23 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
336, 7, 5, 12, 32ucnimalem 24289 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3433fvmpt2 7027 . . . . . . . . . . . . . . . . . . . . 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 8053 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 ∈ (𝑋 × 𝑋) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
3830, 37syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
3936, 38eqtr3d 2779 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
40 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑥 ∈ V
41 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 ∈ V
4240, 41opth 5481 . . . . . . . . . . . . . . . . . . . . . . . 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 4881 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
4935, 48eqtr4d 2780 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
5049eleq1d 2826 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊))
51 df-br 5144 . . . . . . . . . . . . . . . . . 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 3170 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5655reximdv 3170 . . . . . . . . . . . . 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 3140 . . . . . . . . . 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 3151 . . . . . . . . . . 11 (∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6261rexlimivw 3151 . . . . . . . . . 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 841 . . . . . . 7 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6665ralrimiva 3146 . . . . . 6 (((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
6766ex 412 . . . . 5 ((𝜑𝑟𝑈) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
6867reximdva 3168 . . . 4 (𝜑 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
6913, 68mpd 15 . . 3 (𝜑 → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
7032mpofun 7557 . . . . . 6 Fun 𝐺
71 opex 5469 . . . . . . . 8 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
7232, 71dmmpo 8096 . . . . . . 7 dom 𝐺 = (𝑋 × 𝑋)
7317, 72sseqtrrdi 4025 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
74 funimass4 6973 . . . . . 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 3146 . . 3 (𝜑 → ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
78 r19.29r 3116 . . 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 3084 . 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 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  cop 4632   class class class wbr 5143   × cxp 5683  dom cdm 5685  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  UnifOncust 24208   Cnucucn 24284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ust 24209  df-ucn 24285
This theorem is referenced by:  ucnprima  24291
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