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Theorem restutop 24186
Description: Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)
Assertion
Ref Expression
restutop ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))

Proof of Theorem restutop
Dummy variables 𝑎 𝑏 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 481 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋))
2 fvexd 6911 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (unifTop‘𝑈) ∈ V)
3 elfvex 6934 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
43adantr 479 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
5 simpr 483 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
64, 5ssexd 5325 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
7 elrest 17412 . . . . . . 7 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ V) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
82, 6, 7syl2anc 582 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
98biimpa 475 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
10 inss2 4228 . . . . . . 7 (𝑎𝐴) ⊆ 𝐴
11 sseq1 4002 . . . . . . 7 (𝑏 = (𝑎𝐴) → (𝑏𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
1210, 11mpbiri 257 . . . . . 6 (𝑏 = (𝑎𝐴) → 𝑏𝐴)
1312rexlimivw 3140 . . . . 5 (∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴) → 𝑏𝐴)
149, 13syl 17 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏𝐴)
15 simp-5l 783 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑈 ∈ (UnifOn‘𝑋))
1615ad2antrr 724 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
176ad6antr 734 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝐴 ∈ V)
1817, 17xpexd 7754 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝐴 × 𝐴) ∈ V)
19 simplr 767 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑢𝑈)
20 elrestr 17413 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑢𝑈) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2116, 18, 19, 20syl3anc 1368 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
22 inss1 4227 . . . . . . . . . . . . 13 (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢
23 imass1 6106 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥})
25 sstr 3985 . . . . . . . . . . . 12 ((((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
2624, 25mpan 688 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
27 imassrn 6075 . . . . . . . . . . . . . . 15 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝑢 ∩ (𝐴 × 𝐴))
28 rnin 6153 . . . . . . . . . . . . . . 15 ran (𝑢 ∩ (𝐴 × 𝐴)) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
2927, 28sstri 3986 . . . . . . . . . . . . . 14 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
30 inss2 4228 . . . . . . . . . . . . . 14 (ran 𝑢 ∩ ran (𝐴 × 𝐴)) ⊆ ran (𝐴 × 𝐴)
3129, 30sstri 3986 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝐴 × 𝐴)
32 rnxpid 6179 . . . . . . . . . . . . 13 ran (𝐴 × 𝐴) = 𝐴
3331, 32sseqtri 4013 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴
3433a1i 11 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴)
3526, 34ssind 4231 . . . . . . . . . 10 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎𝐴))
3635adantl 480 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎𝐴))
37 simpllr 774 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑏 = (𝑎𝐴))
3836, 37sseqtrrd 4018 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏)
39 imaeq1 6059 . . . . . . . . . 10 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → (𝑣 “ {𝑥}) = ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}))
4039sseq1d 4008 . . . . . . . . 9 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏))
4140rspcev 3606 . . . . . . . 8 (((𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
4221, 38, 41syl2anc 582 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
43 simplr 767 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑎 ∈ (unifTop‘𝑈))
44 simpllr 774 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑏)
45 simpr 483 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑏 = (𝑎𝐴))
4644, 45eleqtrd 2827 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥 ∈ (𝑎𝐴))
4746elin1d 4196 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑎)
48 elutop 24182 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑎 ∈ (unifTop‘𝑈) ↔ (𝑎𝑋 ∧ ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)))
4948simplbda 498 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) → ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5049r19.21bi 3238 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑥𝑎) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5115, 43, 47, 50syl21anc 836 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5242, 51r19.29a 3151 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
539adantr 479 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
5452, 53r19.29a 3151 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
5554ralrimiva 3135 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
56 trust 24178 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
57 elutop 24182 . . . . . 6 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
5856, 57syl 17 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
5958biimpar 476 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
601, 14, 55, 59syl12anc 835 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
6160ex 411 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))))
6261ssrdv 3982 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944  {csn 4630   × cxp 5676  ran crn 5679  cima 5681  cfv 6549  (class class class)co 7419  t crest 17405  UnifOncust 24148  unifTopcutop 24179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-rest 17407  df-ust 24149  df-utop 24180
This theorem is referenced by:  restutopopn  24187
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