Theorem ucnima 23778

Description: An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Hypotheses

Ref	Expression
ucnprima.1	⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
ucnprima.2	⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
ucnprima.3	⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
ucnprima.4	⊢ (𝜑 → 𝑊 ∈ 𝑉)
ucnprima.5	⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)

Assertion

Ref	Expression
ucnima	⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 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.

Step	Hyp	Ref	Expression
1		breq 5150	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝐹‘𝑥)𝑤(𝐹‘𝑦) ↔ (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
2	1	imbi2d 341	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
3	2	ralbidv 3178	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
4	3	rexralbidv 3221	. . . . 5 ⊢ (𝑤 = 𝑊 → (∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)) ↔ ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
5		ucnprima.3	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
6		ucnprima.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
7		ucnprima.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
8		isucn 23775	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑤 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)))))
9	6, 7, 8	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑤 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)))))
10	5, 9	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑤 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦))))
11	10	simprd 497	. . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)))
12		ucnprima.4	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
13	4, 11, 12	rspcdva 3614	. . . 4 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
14		simplll 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → 𝜑)
15		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
16		ustssxp 23701	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
17	6, 16	sylan 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
18	17	sselda 3982	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ 𝑝 ∈ 𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
19	18	adantlr 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
20		simpr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → 𝑝 ∈ 𝑟)
21		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
22		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → 𝑝 ∈ (𝑋 × 𝑋))
23		elxp2 5700	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
24	22, 23	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
25		simpr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
26	25	eleq1d 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝 ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
27	26	adantlr 714	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝 ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
28		df-br 5149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
29	27, 28	bitr4di 289	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝 ∈ 𝑟 ↔ 𝑥𝑟𝑦))
30		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 ∈ (𝑋 × 𝑋))
31		opex 5464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ ∈ V
32		ucnprima.5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
33	6, 7, 5, 12, 32	ucnimalem 23777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
34	33	fvmpt2 7007	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ∈ (𝑋 × 𝑋) ∧ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ ∈ V) → (𝐺‘𝑝) = ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
35	30, 31, 34	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺‘𝑝) = ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
36		simpr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
37		1st2nd2 8011	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 ∈ (𝑋 × 𝑋) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
38	30, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
39	36, 38	eqtr3d 2775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
40		vex 3479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑥 ∈ V
41		vex 3479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
42	40, 41	opth 5476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ (𝑥 = (1st ‘𝑝) ∧ 𝑦 = (2nd ‘𝑝)))
43	39, 42	sylib 217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑥 = (1st ‘𝑝) ∧ 𝑦 = (2nd ‘𝑝)))
44	43	simpld 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑥 = (1st ‘𝑝))
45	44	fveq2d 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑥) = (𝐹‘(1st ‘𝑝)))
46	43	simprd 497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑦 = (2nd ‘𝑝))
47	46	fveq2d 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑦) = (𝐹‘(2nd ‘𝑝)))
48	45, 47	opeq12d 4881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
49	35, 48	eqtr4d 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺‘𝑝) = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
50	49	eleq1d 2819	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑝) ∈ 𝑊 ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ 𝑊))
51		df-br 5149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)𝑊(𝐹‘𝑦) ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ 𝑊)
52	50, 51	bitr4di 289	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑝) ∈ 𝑊 ↔ (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
53	29, 52	imbi12d 345	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊) ↔ (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
54	53	exbiri 810	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))))
55	54	reximdv 3171	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))))
56	55	reximdv 3171	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))))
57	24, 56	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊)))
58	57	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊)))
59	21, 58	r19.29d2r 3141	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))))
60		pm3.35 802	. . . . . . . . . . . 12 ⊢ (((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))
61	60	rexlimivw 3152	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))
62	61	rexlimivw 3152	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))
63	59, 62	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))
64	63	imp 408	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 ∈ 𝑟) → (𝐺‘𝑝) ∈ 𝑊)
65	14, 15, 19, 20, 64	syl1111anc 839	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → (𝐺‘𝑝) ∈ 𝑊)
66	65	ralrimiva 3147	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) → ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊)
67	66	ex 414	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
68	67	reximdva 3169	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑈 ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
69	13, 68	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊)
70	32	mpofun 7529	. . . . . 6 ⊢ Fun 𝐺
71		opex 5464	. . . . . . . 8 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
72	32, 71	dmmpo 8054	. . . . . . 7 ⊢ dom 𝐺 = (𝑋 × 𝑋)
73	17, 72	sseqtrrdi 4033	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺)
74		funimass4 6954	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
75	70, 73, 74	sylancr 588	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
76	75	biimprd 247	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊))
77	76	ralrimiva 3147	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊))
78		r19.29r 3117	. . 3 ⊢ ((∃𝑟 ∈ 𝑈 ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ ∀𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)) → ∃𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)))
79	69, 77, 78	syl2anc 585	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)))
80		pm3.35 802	. . 3 ⊢ ((∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)) → (𝐺 “ 𝑟) ⊆ 𝑊)
81	80	reximi 3085	. 2 ⊢ (∃𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)) → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)
82	79, 81	syl 17	1 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3948  ⟨cop 4634   class class class wbr 5148   × cxp 5674  dom cdm 5676   “ cima 5679  Fun wfun 6535  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406   ∈ cmpo 7408  1^st c1st 7970  2^nd c2nd 7971  UnifOncust 23696   Cn_ucucn 23772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-map 8819  df-ust 23697  df-ucn 23773

This theorem is referenced by:  ucnprima  23779