Theorem restutopopn 24267

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 24262	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
2	1	simprbda 501	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴 ⊆ 𝑋)
3		restutop 24266	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
4	2, 3	syldan 599	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
5		trust 24258	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
6	2, 5	syldan 599	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7		elutop 24262	. . . . . . . 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 501	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ⊆ 𝐴)
10	2	adantr 483	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐴 ⊆ 𝑋)
11	9, 10	sstrd 3937	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ⊆ 𝑋)
12		simp-9l 800	. . . . . . . . . . 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 776	. . . . . . . . . . 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 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) → 𝑤 ∈ 𝑈)
15		ustincl 24237	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈) → (𝑡 ∩ 𝑤) ∈ 𝑈)
16	12, 13, 14, 15	syl3anc 1382	. . . . . . . . . 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 6126	. . . . . . . . . . 11 ⊢ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18		ssrin 4184	. . . . . . . . . . . . . 14 ⊢ ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
19	18	adantl 484	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . 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 6034	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 6127	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
27	26	elv 3449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
28		xpimasn 6156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
29	28	ineq2d 4163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
30	27, 29	eqtrid 2799	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
31		incom 4152	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
32	30, 31	eqtrdi 2803	. . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . 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 3964	. . . . . . . . . . . 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 793	. . . . . . . . . . . 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 3937	. . . . . . . . . . 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 3938	. . . . . . . . . 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 6030	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑡 ∩ 𝑤) → (𝑣 “ {𝑥}) = ((𝑡 ∩ 𝑤) “ {𝑥}))
40	39	sseq1d 3958	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑡 ∩ 𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏))
41	40	rspcev 3572	. . . . . . . . . 10 ⊢ (((𝑡 ∩ 𝑤) ∈ 𝑈 ∧ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
42	16, 38, 41	syl2anc 592	. . . . . . . . 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 790	. . . . . . . . . . 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 734	. . . . . . . . . 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 502	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥 ∈ 𝐴 ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
46	45	r19.21bi 3244	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥 ∈ 𝐴) → ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
47	44, 24, 46	syl2anc 592	. . . . . . . . 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 3160	. . . . . . . 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 776	. . . . . . . . 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 7723	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
51		elrest 17428	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
52	50, 51	sylan2 601	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
53	52	biimpa 479	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
54	43, 49, 53	syl2anc 592	. . . . . . . 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 3160	. . . . . . 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 502	. . . . . . . 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 3244	. . . . . . 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 3160	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
59	58	ralrimiva 3144	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
60		elutop 24262	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
61	60	ad2antrr 734	. . . . 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 721	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ∈ (unifTop‘𝑈))
63		dfss2 3913	. . . . . 6 ⊢ (𝑏 ⊆ 𝐴 ↔ (𝑏 ∩ 𝐴) = 𝑏)
64	9, 63	sylib 220	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∩ 𝐴) = 𝑏)
65	64	eqcomd 2758	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 = (𝑏 ∩ 𝐴))
66		ineq1 4156	. . . . 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 592	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
69		fvex 6865	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
70		elrest 17428	. . . . 5 ⊢ (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
71	69, 70	mpan 698	. . . 4 ⊢ (𝐴 ∈ (unifTop‘𝑈) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
72	71	ad2antlr 735	. . 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 259	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴))
74	4, 73	eqelssd 3948	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  ∀wral 3066  ∃wrex 3076  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895  {csn 4572   × cxp 5634   “ cima 5639  ‘cfv 6506  (class class class)co 7381   ↾_t crest 17421  UnifOncust 24229  unifTopcutop 24259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-rest 17423  df-ust 24230  df-utop 24260

This theorem is referenced by:  ressusp  24293