isucn2 - Metamath Proof Explorer

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Theorem isucn2 22454

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 22453	. . 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 581	. 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 4876	. . . . . . . . . 10 ⊢ (𝑣 = 𝑠 → ((𝐹‘𝑥)𝑣(𝐹‘𝑦) ↔ (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
6	5	imbi2d 332	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
7	6	ralbidv 3196	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → (∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
8	7	rexralbidv 3269	. . . . . . 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 787	. . . . . . 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 22047	. . . . . . . . . . . 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	syl6sseqr 3878	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
15	14	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝑆 ⊆ 𝑉)
16	15	adantr 474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))) → 𝑆 ⊆ 𝑉)
17	16	sselda 3828	. . . . . . 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 3533	. . . . . 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 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
20		isucn2.u	. . . . . . . . . . . 12 ⊢ 𝑈 = ((𝑋 × 𝑋)filGen𝑅)
21	19, 20	syl6eleq 2917	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅))
22		isucn2.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (fBas‘(𝑋 × 𝑋)))
23		elfg 22046	. . . . . . . . . . . . 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 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅)) → ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)
26	21, 25	syldan 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑟 ∈ 𝑅 𝑟 ⊆ 𝑢)
27		ssbr 4918	. . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
30	29	ralrimivw 3177	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
31	30	ralrimivw 3177	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑅) ∧ 𝑟 ⊆ 𝑢) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
32		ralim 3158	. . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . 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 3158	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . 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 3226	. . . . . . . . . . 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 474	. . . . . . . . . 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 3298	. . . . . . . . 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 3241	. . . . . . 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 723	. . . . . 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 3176	. . . 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 22047	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
47	22, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
48	47, 20	syl6sseqr 3878	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
49		ssrexv 3893	. . . . . . . . . 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 4876	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑢 → (𝑥𝑟𝑦 ↔ 𝑥𝑢𝑦))
51	50	imbi1d 333	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → ((𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
52	51	2ralbidv 3199	. . . . . . . . . . 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	cbvrexv 3385	. . . . . . . . . 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 243	. . . . . . . . 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 3173	. . . . . . 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 474	. . . . . 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 2015	. . . . . . . . . . 11 ⊢ Ⅎ𝑠(𝜑 ∧ 𝐹:𝑋⟶𝑌)
59		nfra1 3151	. . . . . . . . . . 11 ⊢ Ⅎ𝑠∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))
60	58, 59	nfan 2004	. . . . . . . . . 10 ⊢ Ⅎ𝑠((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
61		nfv 2015	. . . . . . . . . 10 ⊢ Ⅎ𝑠 𝑣 ∈ 𝑉
62	60, 61	nfan 2004	. . . . . . . . 9 ⊢ Ⅎ𝑠(((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉)
63		simp-4r 805	. . . . . . . . . . 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 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
64		simplr 787	. . . . . . . . . . 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)
65		rspa 3140	. . . . . . . . . . 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)))
66	63, 64, 65	syl2anc 581	. . . . . . . . . 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)))
67		simp-4l 803	. . . . . . . . . . 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))
68		simpr 479	. . . . . . . . . . 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)
69		ssbr 4918	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ⊆ 𝑣 → ((𝐹‘𝑥)𝑠(𝐹‘𝑦) → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
70	69	adantl 475	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ((𝐹‘𝑥)𝑠(𝐹‘𝑦) → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))
71	70	imim2d 57	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑠 ∈ 𝑆) ∧ 𝑠 ⊆ 𝑣) → ((𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)) → (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦))))
72	71	ralimdv 3173	. . . . . . . . . . . . 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))))
73	72	ralimdv 3173	. . . . . . . . . . . 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))))
74	73	reximdv 3225	. . . . . . . . . . 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))))
75	67, 64, 68, 74	syl21anc 873	. . . . . . . . . 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))))
76	66, 75	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)))
77	10	ad3antrrr 723	. . . . . . . . . 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)))
78		simpr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶𝑌) ∧ ∀𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑢𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
79	78, 13	syl6eleq 2917	. . . . . . . . . 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))
80		elfg 22046	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → (𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆) ↔ (𝑣 ⊆ (𝑌 × 𝑌) ∧ ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)))
81	80	simplbda 495	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (fBas‘(𝑌 × 𝑌)) ∧ 𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆)) → ∃𝑠 ∈ 𝑆 𝑠 ⊆ 𝑣)
82	77, 79, 81	syl2anc 581	. . . . . . . . 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)
83	62, 76, 82	r19.29af 3287	. . . . . . . 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)))
84	83	ralrimiva 3176	. . . . . . 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)))
85	84	ex 403	. . . . . 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))))
86	57, 85	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))))
87	86	imp 397	. . . 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)))
88	45, 87	impbida 837	. . 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))))
89	88	pm5.32da 576	. 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)))))
90	4, 89	bitrd 271	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 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∃wrex 3119 ⊆ wss 3799 class class class wbr 4874 × cxp 5341 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 fBascfbas 20095 filGencfg 20096 UnifOncust 22374 Cn_ucucn 22450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-map 8125 df-fbas 20104 df-fg 20105 df-ust 22375 df-ucn 22451

This theorem is referenced by: metucn 22747

W3C validator