Theorem restutopopn 21852

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 21847	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
2	1	simprbda 651	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴 ⊆ 𝑋)
3		restutop 21851	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
4	2, 3	syldan 486	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
5		trust 21843	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
6	2, 5	syldan 486	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7		elutop 21847	. . . . . . . . . 10 ⊢ ((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	. . . . . . . . 9 ⊢ ((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 651	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ⊆ 𝐴)
10	2	adantr 480	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐴 ⊆ 𝑋)
11	9, 10	sstrd 3578	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ⊆ 𝑋)
12		simp-9l 812	. . . . . . . . . . . . 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 ∈ (UnifOn‘𝑋))
13		simplr 788	. . . . . . . . . . . . 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 ∈ 𝑈)
14		simp-4r 803	. . . . . . . . . . . . 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) → 𝑤 ∈ 𝑈)
15		ustincl 21821	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈) → (𝑡 ∩ 𝑤) ∈ 𝑈)
16	12, 13, 14, 15	syl3anc 1318	. . . . . . . . . . . 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 ∩ 𝑤) ∈ 𝑈)
17		inimass 5468	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18		ssrin 3800	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
19	18	adantl 481	. . . . . . . . . . . . . . 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) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
20		simpllr 795	. . . . . . . . . . . . . . . . 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))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
21	20	imaeq1d 5384	. . . . . . . . . . . . . . . 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))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) = ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}))
22	9	ad5antr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((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 805	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((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 3569	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((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 758	. . . . . . . . . . . . . . . . 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))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑥 ∈ 𝐴)
26		vex 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
27		inimasn 5469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
28	26, 27	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
29		xpimasn 5498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
30	29	ineq2d 3776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
31	28, 30	syl5eq 2656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
32		incom 3767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
33	31, 32	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
34	25, 33	syl 17	. . . . . . . . . . . . . . . 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))) ∧ 𝑡 ∈ 𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
35	21, 34	eqtrd 2644	. . . . . . . . . . . . . . 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 “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
36	19, 35	sseqtr4d 3605	. . . . . . . . . . . . . 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) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝑢 “ {𝑥}))
37		simp-5r 805	. . . . . . . . . . . . . 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}) ⊆ 𝑏)
38	36, 37	sstrd 3578	. . . . . . . . . . . . 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})) ⊆ 𝑏)
39	17, 38	syl5ss 3579	. . . . . . . . . . . 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 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏)
40		imaeq1 5380	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑡 ∩ 𝑤) → (𝑣 “ {𝑥}) = ((𝑡 ∩ 𝑤) “ {𝑥}))
41	40	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑡 ∩ 𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏))
42	41	rspcev 3282	. . . . . . . . . . . 12 ⊢ (((𝑡 ∩ 𝑤) ∈ 𝑈 ∧ ((𝑡 ∩ 𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
43	16, 39, 42	syl2anc 691	. . . . . . . . . . 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) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
44		simp-4l 802	. . . . . . . . . . . . 13 ⊢ ((((((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)))
45	44	ad2antrr 758	. . . . . . . . . . . 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))) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)))
46	1	simplbda 652	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥 ∈ 𝐴 ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
47	46	r19.21bi 2916	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥 ∈ 𝐴) → ∃𝑡 ∈ 𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
48	45, 24, 47	syl2anc 691	. . . . . . . . . . 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)
49	43, 48	r19.29a 3060	. . . . . . . . . 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))) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
50		simplr 788	. . . . . . . . . . 11 ⊢ ((((((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)))
51		sqxpexg 6861	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
52		elrest 15911	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
53	51, 52	sylan2 490	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
54	53	biimpa 500	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑤 ∈ 𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
55	44, 50, 54	syl2anc 691	. . . . . . . . . 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)))
56	49, 55	r19.29a 3060	. . . . . . . . 9 ⊢ ((((((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)
57	8	simplbda 652	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑥 ∈ 𝑏 ∃𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
58	57	r19.21bi 2916	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) → ∃𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
59	56, 58	r19.29a 3060	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑥 ∈ 𝑏) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
60	59	ralrimiva 2949	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
61		elutop 21847	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
62	61	ad2antrr 758	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
63	11, 60, 62	mpbir2and 959	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ∈ (unifTop‘𝑈))
64		df-ss 3554	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝐴 ↔ (𝑏 ∩ 𝐴) = 𝑏)
65	9, 64	sylib 207	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∩ 𝐴) = 𝑏)
66	65	eqcomd 2616	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 = (𝑏 ∩ 𝐴))
67		ineq1 3769	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 ∩ 𝐴) = (𝑏 ∩ 𝐴))
68	67	eqeq2d 2620	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑏 = (𝑎 ∩ 𝐴) ↔ 𝑏 = (𝑏 ∩ 𝐴)))
69	68	rspcev 3282	. . . . . 6 ⊢ ((𝑏 ∈ (unifTop‘𝑈) ∧ 𝑏 = (𝑏 ∩ 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
70	63, 66, 69	syl2anc 691	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
71		fvex 6113	. . . . . . 7 ⊢ (unifTop‘𝑈) ∈ V
72		elrest 15911	. . . . . . 7 ⊢ (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
73	71, 72	mpan 702	. . . . . 6 ⊢ (𝐴 ∈ (unifTop‘𝑈) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
74	73	ad2antlr 759	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
75	70, 74	mpbird 246	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴))
76	75	ex 449	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)))
77	76	ssrdv 3574	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ⊆ ((unifTop‘𝑈) ↾t 𝐴))
78	4, 77	eqssd 3585	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125   × cxp 5036   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  UnifOncust 21813  unifTopcutop 21844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-ust 21814  df-utop 21845

This theorem is referenced by:  ressusp  21879