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Theorem ucnima 22008
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 ucnprima.4 . . . . 5 (𝜑𝑊𝑉)
2 ucnprima.3 . . . . . . 7 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
3 ucnprima.1 . . . . . . . 8 (𝜑𝑈 ∈ (UnifOn‘𝑋))
4 ucnprima.2 . . . . . . . 8 (𝜑𝑉 ∈ (UnifOn‘𝑌))
5 isucn 22005 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))))
63, 4, 5syl2anc 692 . . . . . . 7 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))))
72, 6mpbid 222 . . . . . 6 (𝜑 → (𝐹:𝑋𝑌 ∧ ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦))))
87simprd 479 . . . . 5 (𝜑 → ∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)))
9 breq 4620 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝐹𝑥)𝑤(𝐹𝑦) ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
109imbi2d 330 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
1110ralbidv 2981 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
1211rexralbidv 3052 . . . . . 6 (𝑤 = 𝑊 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) ↔ ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
1312rspcv 3294 . . . . 5 (𝑊𝑉 → (∀𝑤𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑤(𝐹𝑦)) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
141, 8, 13sylc 65 . . . 4 (𝜑 → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
15 simplll 797 . . . . . . . . 9 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝜑)
16 simplr 791 . . . . . . . . 9 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
1715, 16jca 554 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → (𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
18 ustssxp 21931 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
193, 18sylan 488 . . . . . . . . . 10 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
2019sselda 3587 . . . . . . . . 9 (((𝜑𝑟𝑈) ∧ 𝑝𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
2120adantlr 750 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
22 simpr 477 . . . . . . . 8 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → 𝑝𝑟)
23 simplr 791 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)))
24 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → 𝑝 ∈ (𝑋 × 𝑋))
25 elxp2 5097 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝑋 × 𝑋) ↔ ∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
2624, 25sylib 208 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
27 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
2827eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
2928adantlr 750 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
30 df-br 4619 . . . . . . . . . . . . . . . . . 18 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
3129, 30syl6bbr 278 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝𝑟𝑥𝑟𝑦))
32 simplr 791 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 ∈ (𝑋 × 𝑋))
33 opex 4898 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V
34 ucnprima.5 . . . . . . . . . . . . . . . . . . . . . . 23 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
353, 4, 2, 1, 34ucnimalem 22007 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3635fvmpt2 6253 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ (𝑋 × 𝑋) ∧ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ ∈ V) → (𝐺𝑝) = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
3732, 33, 36sylancl 693 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
38 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
39 1st2nd2 7157 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 ∈ (𝑋 × 𝑋) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
4032, 39syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
4138, 40eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
42 vex 3192 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑥 ∈ V
43 vex 3192 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 ∈ V
4442, 43opth 4910 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑥, 𝑦⟩ = ⟨(1st𝑝), (2nd𝑝)⟩ ↔ (𝑥 = (1st𝑝) ∧ 𝑦 = (2nd𝑝)))
4541, 44sylib 208 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑥 = (1st𝑝) ∧ 𝑦 = (2nd𝑝)))
4645simpld 475 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑥 = (1st𝑝))
4746fveq2d 6157 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹𝑥) = (𝐹‘(1st𝑝)))
4845simprd 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑦 = (2nd𝑝))
4948fveq2d 6157 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹𝑦) = (𝐹‘(2nd𝑝)))
5047, 49opeq12d 4383 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
5137, 50eqtr4d 2658 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺𝑝) = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
5251eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊))
53 df-br 4619 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥)𝑊(𝐹𝑦) ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ 𝑊)
5452, 53syl6bbr 278 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺𝑝) ∈ 𝑊 ↔ (𝐹𝑥)𝑊(𝐹𝑦)))
5531, 54imbi12d 334 . . . . . . . . . . . . . . . 16 (((𝜑𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝑝𝑟 → (𝐺𝑝) ∈ 𝑊) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))))
5655exbiri 651 . . . . . . . . . . . . . . 15 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5756reximdv 3011 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5857reximdv 3011 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑥𝑋𝑦𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
5926, 58mpd 15 . . . . . . . . . . . 12 ((𝜑𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊)))
6059adantlr 750 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊)))
6123, 60r19.29d2r 3073 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))))
62 pm3.35 610 . . . . . . . . . . . 12 (((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6362rexlimivw 3023 . . . . . . . . . . 11 (∃𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6463rexlimivw 3023 . . . . . . . . . 10 (∃𝑥𝑋𝑦𝑋 ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6561, 64syl 17 . . . . . . . . 9 (((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝𝑟 → (𝐺𝑝) ∈ 𝑊))
6665imp 445 . . . . . . . 8 ((((𝜑 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6717, 21, 22, 66syl21anc 1322 . . . . . . 7 ((((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) ∧ 𝑝𝑟) → (𝐺𝑝) ∈ 𝑊)
6867ralrimiva 2961 . . . . . 6 (((𝜑𝑟𝑈) ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦))) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
6968ex 450 . . . . 5 ((𝜑𝑟𝑈) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7069reximdva 3012 . . . 4 (𝜑 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑊(𝐹𝑦)) → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7114, 70mpd 15 . . 3 (𝜑 → ∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊)
7234mpt2fun 6722 . . . . . 6 Fun 𝐺
73 opex 4898 . . . . . . . 8 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
7434, 73dmmpt2 7192 . . . . . . 7 dom 𝐺 = (𝑋 × 𝑋)
7519, 74syl6sseqr 3636 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
76 funimass4 6209 . . . . . 6 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊 ↔ ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7772, 75, 76sylancr 694 . . . . 5 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊 ↔ ∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊))
7877biimprd 238 . . . 4 ((𝜑𝑟𝑈) → (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
7978ralrimiva 2961 . . 3 (𝜑 → ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊))
80 r19.29r 3067 . . 3 ((∃𝑟𝑈𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ ∀𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → ∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)))
8171, 79, 80syl2anc 692 . 2 (𝜑 → ∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)))
82 pm3.35 610 . . 3 ((∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → (𝐺𝑟) ⊆ 𝑊)
8382reximi 3006 . 2 (∃𝑟𝑈 (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 ∧ (∀𝑝𝑟 (𝐺𝑝) ∈ 𝑊 → (𝐺𝑟) ⊆ 𝑊)) → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
8481, 83syl 17 1 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  wss 3559  cop 4159   class class class wbr 4618   × cxp 5077  dom cdm 5079  cima 5082  Fun wfun 5846  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  1st c1st 7118  2nd c2nd 7119  UnifOncust 21926   Cnucucn 22002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811  df-ust 21927  df-ucn 22003
This theorem is referenced by:  ucnprima  22009
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