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Theorem isucn2 24446

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 24445	. . 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 595	. 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 5110	. . . . . . . . . 10 ⊢ (𝑣 = 𝑠 → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
6	5	imbi2d 343	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
7	6	ralbidv 3187	. . . . . . . 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 780	. . . . . . 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 24040	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → 𝑆 ⊆ ((𝑌 × 𝑌)filGen𝑆))
12	10, 11	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ ((𝑌 × 𝑌)filGen𝑆))
13		isucn2.v	. . . . . . . . . . 11 ⊢ 𝑉 = ((𝑌 × 𝑌)filGen𝑆)
14	12, 13	sseqtrrdi 3977	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
15	14	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝑆 ⊆ 𝑉)
16	15	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) → 𝑆 ⊆ 𝑉)
17	16	sselda 3936	. . . . . . 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 3581	. . . . . 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 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
20		isucn2.u	. . . . . . . . . . . 12 ⊢ 𝑈 = ((𝑋 × 𝑋)filGen𝑅)
21	19, 20	eleqtrdi 2872	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅))
22		isucn2.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (fBas‘(𝑋 × 𝑋)))
23		elfg 24039	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅) ↔ (𝑢 ⊆ (𝑋 × 𝑋) ∧ ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)))
24	22, 23	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅) ↔ (𝑢 ⊆ (𝑋 × 𝑋) ∧ ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)))
25	24	simplbda 504	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅)) → ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)
26	21, 25	syldan 602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)
27		ssbr 5154	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ⊆ 𝑢 → (𝑥𝑟𝑦 → 𝑥𝑢𝑦))
28	27	imim1d 83	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ⊆ 𝑢 → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
29	28	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
30	29	ralrimivw 3160	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
31	30	ralrimivw 3160	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
32		ralim 3104	. . . . . . . . . . . . . . 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 3101	. . . . . . . . . . . . . 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 3104	. . . . . . . . . . . . . 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 19	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 3177	. . . . . . . . . . 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 485	. . . . . . . . . 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 16	. . . . . . . . 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 3190	. . . . . . . . 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 18	. . . . . . . 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 3165	. . . . . . 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 742	. . . . . 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 16	. . . . 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 3156	. . . 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 24040	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
47	22, 46	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
48	47, 20	sseqtrrdi 3977	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
49		ssrexv 4006	. . . . . . . . . 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 5110	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑢 → (𝑥𝑟𝑦 ↔ 𝑥𝑢𝑦))
51	50	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
52	51	2ralbidv 3228	. . . . . . . . . . 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 3243	. . . . . . . . . 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	imbitrdi 254	. . . . . . . . 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 18	. . . . . . . 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 3178	. . . . . . 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 485	. . . . . 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 1943	. . . . . . . . . . 11 ⊢ Ⅎ𝑠(𝜑 ∧ 𝐹:𝑋⟶𝑌)
59		nfra1 3288	. . . . . . . . . . 11 ⊢ Ⅎ𝑠∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))
60	58, 59	nfan 1928	. . . . . . . . . 10 ⊢ Ⅎ𝑠((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
61		nfv 1943	. . . . . . . . . 10 ⊢ Ⅎ𝑠 𝑣 ∈ 𝑉
62	60, 61	nfan 1928	. . . . . . . . 9 ⊢ Ⅎ𝑠(((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉)
63		rspa 3253	. . . . . . . . . . 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 772	. . . . . . . . . 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 794	. . . . . . . . . . 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 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) → 𝑠 ∈ 𝑆)
67		simpr 489	. . . . . . . . . . 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 5154	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ⊆ 𝑣 → ((𝐹‘𝑥)𝑠(𝐹‘𝑦) → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
69	68	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ((𝐹‘𝑥)𝑠(𝐹‘𝑦) → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
70	69	imim2d 58	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
71	70	ralimdv 3178	. . . . . . . . . . . . 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 3178	. . . . . . . . . . . 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 3179	. . . . . . . . . . 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 850	. . . . . . . . . 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 16	. . . . . . . . 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 742	. . . . . . . . . 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 489	. . . . . . . . . . 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 2872	. . . . . . . . . 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 24039	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → (𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆) ↔ (𝑣 ⊆ (𝑌 × 𝑌) ∧ ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)))
80	79	simplbda 504	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (fBas‘(𝑌 × 𝑌)) ∧ 𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆)) → ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)
81	76, 78, 80	syl2anc 595	. . . . . . . . 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 3273	. . . . . . . 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 3156	. . . . . . 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 417	. . . . . 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 48	. . . . 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 411	. . . 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 812	. . 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 589	. 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 282	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 ⊆ wss 3904 class class class wbr 5108 × cxp 5658 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 fBascfbas 21521 filGencfg 21522 UnifOncust 24368 Cn_ucucn 24442

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 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-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8824 df-fbas 21530 df-fg 21531 df-ust 24369 df-ucn 24443

This theorem is used by: metucn 24739

W3C validator