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Theorem isucn2 23431

Description: The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉", expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.)

**Hypotheses**
Ref	Expression
isucn2.u	⊢ 𝑈 = ((𝑋 × 𝑋)filGen𝑅)
isucn2.v	⊢ 𝑉 = ((𝑌 × 𝑌)filGen𝑆)
isucn2.1	⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
isucn2.2	⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
isucn2.3	⊢ (𝜑 → 𝑅 ∈ (fBas‘(𝑋 × 𝑋)))
isucn2.4	⊢ (𝜑 → 𝑆 ∈ (fBas‘(𝑌 × 𝑌)))

**Assertion**
Ref	Expression
isucn2	⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))

Distinct variable groups: 𝑠,𝑟,𝑥,𝑦,𝐹 𝑅,𝑟,𝑥,𝑦 𝑆,𝑠,𝑥,𝑦 𝑈,𝑟,𝑠,𝑥,𝑦 𝑉,𝑠,𝑥 𝑋,𝑟,𝑠,𝑥,𝑦 𝑌,𝑠,𝑥,𝑦 𝜑,𝑟,𝑠,𝑥,𝑦 Allowed substitution hints: 𝑅(𝑠) 𝑆(𝑟) 𝑉(𝑦,𝑟) 𝑌(𝑟)

Proof of Theorem isucn2 Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isucn2.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
2		isucn2.2	. . 3 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
3		isucn 23430	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))))
5		breq 5076	. . . . . . . . . 10 ⊢ (𝑣 = 𝑠 → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
6	5	imbi2d 341	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
7	6	ralbidv 3112	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
8	7	rexralbidv 3230	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
9		simplr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ∧ 𝑠 ∈ 𝑆) → ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
10		isucn2.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (fBas‘(𝑌 × 𝑌)))
11		ssfg 23023	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → 𝑆 ⊆ ((𝑌 × 𝑌)filGen𝑆))
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ ((𝑌 × 𝑌)filGen𝑆))
13		isucn2.v	. . . . . . . . . . 11 ⊢ 𝑉 = ((𝑌 × 𝑌)filGen𝑆)
14	12, 13	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
15	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝑆 ⊆ 𝑉)
16	15	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) → 𝑆 ⊆ 𝑉)
17	16	sselda 3921	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑉)
18	8, 9, 17	rspcdva 3562	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ∧ 𝑠 ∈ 𝑆) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
19		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
20		isucn2.u	. . . . . . . . . . . 12 ⊢ 𝑈 = ((𝑋 × 𝑋)filGen𝑅)
21	19, 20	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅))
22		isucn2.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (fBas‘(𝑋 × 𝑋)))
23		elfg 23022	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅) ↔ (𝑢 ⊆ (𝑋 × 𝑋) ∧ ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)))
24	22, 23	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅) ↔ (𝑢 ⊆ (𝑋 × 𝑋) ∧ ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)))
25	24	simplbda 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅)) → ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)
26	21, 25	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)
27		ssbr 5118	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ⊆ 𝑢 → (𝑥𝑟𝑦 → 𝑥𝑢𝑦))
28	27	imim1d 82	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ⊆ 𝑢 → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
29	28	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
30	29	ralrimivw 3104	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
31	30	ralrimivw 3104	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
32		ralim 3083	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) → (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
33	32	ralimi 3087	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) → ∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
34		ralim 3083	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
35	31, 33, 34	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
36	35	ex 413	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑟 ⊆ 𝑢 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
37	36	reximdva 3203	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢 → ∃𝑟 ∈ 𝑅 (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
38	37	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢 → ∃𝑟 ∈ 𝑅 (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
39	26, 38	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑟 ∈ 𝑅 (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
40		r19.37v 3274	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝑅 (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
41	39, 40	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
42	41	rexlimdva 3213	. . . . . . 7 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
43	42	ad3antrrr 727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ∧ 𝑠 ∈ 𝑆) → (∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
44	18, 43	mpd 15	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ∧ 𝑠 ∈ 𝑆) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
45	44	ralrimiva 3103	. . . 4 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) → ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
46		ssfg 23023	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
47	22, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
48	47, 20	sseqtrrdi 3972	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
49		ssrexv 3988	. . . . . . . . . 10 ⊢ (𝑅 ⊆ 𝑈 → (∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
50		breq 5076	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑢 → (𝑥𝑟𝑦 ↔ 𝑥𝑢𝑦))
51	50	imbi1d 342	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
52	51	2ralbidv 3129	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑢 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
53	52	cbvrexvw 3384	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) ↔ ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
54	49, 53	syl6ib 250	. . . . . . . . 9 ⊢ (𝑅 ⊆ 𝑈 → (∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
55	48, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
56	55	ralimdv 3109	. . . . . . 7 ⊢ (𝜑 → (∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
57	56	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
58		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑠(𝜑 ∧ 𝐹:𝑋⟶𝑌)
59		nfra1 3144	. . . . . . . . . . 11 ⊢ Ⅎ𝑠∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))
60	58, 59	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑠((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
61		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑠 𝑣 ∈ 𝑉
62	60, 61	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑠(((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉)
63		rspa 3132	. . . . . . . . . . 11 ⊢ ((∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) ∧ 𝑠 ∈ 𝑆) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
64	63	ad5ant24 758	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
65		simp-4l 780	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → (𝜑 ∧ 𝐹:𝑋⟶𝑌))
66		simplr 766	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → 𝑠 ∈ 𝑆)
67		simpr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → 𝑠 ⊆ 𝑣)
68		ssbr 5118	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ⊆ 𝑣 → ((𝐹‘𝑥)𝑠(𝐹‘𝑦) → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
69	68	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ((𝐹‘𝑥)𝑠(𝐹‘𝑦) → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
70	69	imim2d 57	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
71	70	ralimdv 3109	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
72	71	ralimdv 3109	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
73	72	reximdv 3202	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → (∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
74	65, 66, 67, 73	syl21anc 835	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → (∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
75	64, 74	mpd 15	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
76	10	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) → 𝑆 ∈ (fBas‘(𝑌 × 𝑌)))
77		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
78	77, 13	eleqtrdi 2849	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆))
79		elfg 23022	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → (𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆) ↔ (𝑣 ⊆ (𝑌 × 𝑌) ∧ ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)))
80	79	simplbda 500	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (fBas‘(𝑌 × 𝑌)) ∧ 𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆)) → ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)
81	76, 78, 80	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) → ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)
82	62, 75, 81	r19.29af 3262	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) → ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
83	82	ralrimiva 3103	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) → ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
84	83	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
85	57, 84	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
86	85	imp 407	. . . 4 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) → ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
87	45, 86	impbida 798	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → (∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
88	87	pm5.32da 579	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
89	4, 88	bitrd 278	1 ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cn_u𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 × cxp 5587 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 fBascfbas 20585 filGencfg 20586 UnifOncust 23351 Cn_ucucn 23427

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-fbas 20594 df-fg 20595 df-ust 23352 df-ucn 23428

This theorem is referenced by: metucn 23727

W3C validator