Theorem ucnima 24416

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 5110	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝐹‘𝑥)𝑤(𝐹‘𝑦) ↔ (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
2	1	imbi2d 343	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
3	2	ralbidv 3186	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
4	3	rexralbidv 3229	. . . . 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 24413	. . . . . . . 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 595	. . . . . . 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 235	. . . . . 6 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑤 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑤(𝐹‘𝑦))))
11	10	simprd 500	. . . . 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 3581	. . . 4 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
14		simplll 786	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → 𝜑)
15		simplr 780	. . . . . . . 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 24341	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
17	6, 16	sylan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
18	17	sselda 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ 𝑝 ∈ 𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
19	18	adantlr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → 𝑝 ∈ (𝑋 × 𝑋))
20		simpr 489	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → 𝑝 ∈ 𝑟)
21		simplr 780	. . . . . . . . . . 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		elxp2 5685	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
23	22	bilani 509	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩)
24		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
25	24	eleq1d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝 ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
26	25	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝 ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟))
27		df-br 5109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
28	26, 27	bitr4di 292	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑝 ∈ 𝑟 ↔ 𝑥𝑟𝑦))
29		simplr 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 ∈ (𝑋 × 𝑋))
30		opex 5445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ ∈ V
31		ucnprima.5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
32	6, 7, 5, 12, 31	ucnimalem 24415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
33	32	fvmpt2 7001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ∈ (𝑋 × 𝑋) ∧ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ ∈ V) → (𝐺‘𝑝) = ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
34	29, 30, 33	sylancl 597	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺‘𝑝) = ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
35		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨𝑥, 𝑦⟩)
36		1st2nd2 8024	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 ∈ (𝑋 × 𝑋) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
37	29, 36	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
38	35, 37	eqtr3d 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
39		vex 3457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑥 ∈ V
40		vex 3457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
41	39, 40	opth 5458	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ (𝑥 = (1st ‘𝑝) ∧ 𝑦 = (2nd ‘𝑝)))
42	38, 41	sylib 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝑥 = (1st ‘𝑝) ∧ 𝑦 = (2nd ‘𝑝)))
43	42	simpld 499	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑥 = (1st ‘𝑝))
44	43	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑥) = (𝐹‘(1st ‘𝑝)))
45	42	simprd 500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → 𝑦 = (2nd ‘𝑝))
46	45	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑦) = (𝐹‘(2nd ‘𝑝)))
47	44, 46	opeq12d 4845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
48	34, 47	eqtr4d 2799	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → (𝐺‘𝑝) = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
49	48	eleq1d 2846	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑝) ∈ 𝑊 ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ 𝑊))
50		df-br 5109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)𝑊(𝐹‘𝑦) ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ 𝑊)
51	49, 50	bitr4di 292	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝐺‘𝑝) ∈ 𝑊 ↔ (𝐹‘𝑥)𝑊(𝐹‘𝑦)))
52	28, 51	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊) ↔ (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))))
53	52	exbiri 822	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))))
54	53	reximdv 3178	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (∃𝑦 ∈ 𝑋 𝑝 = ⟨𝑥, 𝑦⟩ → ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))))
55	54	reximdv 3178	. . . . . . . . . . . . 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))))
56	23, 55	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑋 × 𝑋)) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊)))
57	56	adantlr 727	. . . . . . . . . . 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)))
58	21, 57	r19.29d2r 3150	. . . . . . . . . 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))))
59		pm3.35 814	. . . . . . . . . . . 12 ⊢ (((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) ∧ ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))
60	59	rexlimivw 3160	. . . . . . . . . . 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))
61	60	rexlimivw 3160	. . . . . . . . . 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))
62	58, 61	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) → (𝑝 ∈ 𝑟 → (𝐺‘𝑝) ∈ 𝑊))
63	62	imp 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ (𝑋 × 𝑋)) ∧ 𝑝 ∈ 𝑟) → (𝐺‘𝑝) ∈ 𝑊)
64	14, 15, 19, 20, 63	syl1111anc 853	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) ∧ 𝑝 ∈ 𝑟) → (𝐺‘𝑝) ∈ 𝑊)
65	64	ralrimiva 3155	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦))) → ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊)
66	65	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
67	66	reximdva 3176	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑊(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑈 ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
68	13, 67	mpd 16	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊)
69	31	mpofun 7534	. . . . . 6 ⊢ Fun 𝐺
70		opex 5445	. . . . . . . 8 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
71	31, 70	dmmpo 8067	. . . . . . 7 ⊢ dom 𝐺 = (𝑋 × 𝑋)
72	17, 71	sseqtrrdi 3977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺)
73		funimass4 6945	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
74	69, 72, 73	sylancr 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊))
75	74	biimprd 251	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊))
76	75	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊))
77		r19.29r 3127	. . 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)))
78	68, 76, 77	syl2anc 595	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)))
79		pm3.35 814	. . 3 ⊢ ((∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)) → (𝐺 “ 𝑟) ⊆ 𝑊)
80	79	reximi 3101	. 2 ⊢ (∃𝑟 ∈ 𝑈 (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 ∧ (∀𝑝 ∈ 𝑟 (𝐺‘𝑝) ∈ 𝑊 → (𝐺 “ 𝑟) ⊆ 𝑊)) → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)
81	78, 80	syl 18	1 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  Vcvv 3453   ⊆ wss 3904  ⟨cop 4594   class class class wbr 5108   × cxp 5659  dom cdm 5661   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7983  2^nd c2nd 7984  UnifOncust 24336   Cn_ucucn 24410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-map 8825  df-ust 24337  df-ucn 24411

This theorem is referenced by:  ucnprima  24417