Theorem restutopopn 23590

Description: The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
restutopopn	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))

Proof of Theorem restutopopn

Dummy variables 𝑎 𝑏 𝑡 𝑢 𝑤 𝑥 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elutop 23585	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
2	1	simprbda 499	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴 ⊆ 𝑋)
3		restutop 23589	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
4	2, 3	syldan 591	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
5		trust 23581	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
6	2, 5	syldan 591	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7		elutop 23585	. . . . . . . 8 ⊢ ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)))
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)))
9	8	simprbda 499	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ⊆ 𝐴)
10	2	adantr 481	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐴 ⊆ 𝑋)
11	9, 10	sstrd 3954	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ⊆ 𝑋)
12		simp-9l 791	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑈 ∈ (UnifOn‘𝑋))
13		simplr 767	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑡 ∈ 𝑈)
14		simp-4r 782	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑤 ∈ 𝑈)
15		ustincl 23559	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈) → (𝑡 ∩ 𝑤) ∈ 𝑈)
16	12, 13, 14, 15	syl3anc 1371	. . . . . . . . . 10 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑡 ∩ 𝑤) ∈ 𝑈)
17		inimass 6107	. . . . . . . . . . 11 ⊢ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18		ssrin 4193	. . . . . . . . . . . . . 14 ⊢ ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
19	18	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
20		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
21	20	imaeq1d 6012	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) = ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}))
22	9	ad5antr 732	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → 𝑏 ⊆ 𝐴)
23		simp-5r 784	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → 𝑥 ∈ 𝑏)
24	22, 23	sseldd 3945	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → 𝑥 ∈ 𝐴)
25	24	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑥 ∈ 𝐴)
26		inimasn 6108	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
27	26	elv 3451	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
28		xpimasn 6137	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
29	28	ineq2d 4172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
30	27, 29	eqtrid 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
31		incom 4161	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
32	30, 31	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
33	25, 32	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
34	21, 33	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
35	19, 34	sseqtrrd 3985	. . . . . . . . . . . 12 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝑢 “ {𝑥}))
36		simp-5r 784	. . . . . . . . . . . 12 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) ⊆ 𝑏)
37	35, 36	sstrd 3954	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ 𝑏)
38	17, 37	sstrid 3955	. . . . . . . . . 10 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏)
39		imaeq1 6008	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑡 ∩ 𝑤) → (𝑣 “ {𝑥}) = ((𝑡 ∩ 𝑤) “ {𝑥}))
40	39	sseq1d 3975	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑡 ∩ 𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏))
41	40	rspcev 3581	. . . . . . . . . 10 ⊢ (((𝑡 ∩ 𝑤) ∈ 𝑈 ∧ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
42	16, 38, 41	syl2anc 584	. . . . . . . . 9 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
43		simp-4l 781	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)))
44	43	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)))
45	1	simplbda 500	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥 ∈ 𝐴 ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
46	45	r19.21bi 3234	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥 ∈ 𝐴) → ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
47	44, 24, 46	syl2anc 584	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
48	42, 47	r19.29a 3159	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤 ∈ 𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
49		simplr 767	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
50		sqxpexg 7689	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
51		elrest 17309	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
52	50, 51	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
53	52	biimpa 477	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
54	43, 49, 53	syl2anc 584	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
55	48, 54	r19.29a 3159	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
56	8	simplbda 500	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑥 ∈ 𝑏 ∃𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
57	56	r19.21bi 3234	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) → ∃𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
58	55, 57	r19.29a 3159	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
59	58	ralrimiva 3143	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
60		elutop 23585	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
61	60	ad2antrr 724	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
62	11, 59, 61	mpbir2and 711	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ∈ (unifTop‘𝑈))
63		df-ss 3927	. . . . . 6 ⊢ (𝑏 ⊆ 𝐴 ↔ (𝑏 ∩ 𝐴) = 𝑏)
64	9, 63	sylib 217	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∩ 𝐴) = 𝑏)
65	64	eqcomd 2742	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 = (𝑏 ∩ 𝐴))
66		ineq1 4165	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑎 ∩ 𝐴) = (𝑏 ∩ 𝐴))
67	66	rspceeqv 3595	. . . 4 ⊢ ((𝑏 ∈ (unifTop‘𝑈) ∧ 𝑏 = (𝑏 ∩ 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
68	62, 65, 67	syl2anc 584	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
69		fvex 6855	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
70		elrest 17309	. . . . 5 ⊢ (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
71	69, 70	mpan 688	. . . 4 ⊢ (𝐴 ∈ (unifTop‘𝑈) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
72	71	ad2antlr 725	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
73	68, 72	mpbird 256	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴))
74	4, 73	eqelssd 3965	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  {csn 4586   × cxp 5631   “ cima 5636  ‘cfv 6496  (class class class)co 7357   ↾_t crest 17302  UnifOncust 23551  unifTopcutop 23582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-rest 17304  df-ust 23552  df-utop 23583

This theorem is referenced by:  ressusp  23616