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Theorem restutop 22088
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 472 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋))
2 fvexd 6241 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (unifTop‘𝑈) ∈ V)
3 elfvex 6259 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
43adantr 480 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
5 simpr 476 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
64, 5ssexd 4838 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
7 elrest 16135 . . . . . . 7 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ V) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
82, 6, 7syl2anc 694 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
98biimpa 500 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
10 inss2 3867 . . . . . . 7 (𝑎𝐴) ⊆ 𝐴
11 sseq1 3659 . . . . . . 7 (𝑏 = (𝑎𝐴) → (𝑏𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
1210, 11mpbiri 248 . . . . . 6 (𝑏 = (𝑎𝐴) → 𝑏𝐴)
1312rexlimivw 3058 . . . . 5 (∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴) → 𝑏𝐴)
149, 13syl 17 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏𝐴)
15 simp-5l 825 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑈 ∈ (UnifOn‘𝑋))
1615ad2antrr 762 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
176ad6antr 777 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝐴 ∈ V)
18 xpexg 7002 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
1917, 17, 18syl2anc 694 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝐴 × 𝐴) ∈ V)
20 simplr 807 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑢𝑈)
21 elrestr 16136 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑢𝑈) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2216, 19, 20, 21syl3anc 1366 . . . . . . . 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)))
23 inss1 3866 . . . . . . . . . . . . 13 (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢
24 imass1 5535 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}))
2523, 24ax-mp 5 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥})
26 sstr 3644 . . . . . . . . . . . 12 ((((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
2725, 26mpan 706 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
28 imassrn 5512 . . . . . . . . . . . . . . 15 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝑢 ∩ (𝐴 × 𝐴))
29 rnin 5577 . . . . . . . . . . . . . . 15 ran (𝑢 ∩ (𝐴 × 𝐴)) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
3028, 29sstri 3645 . . . . . . . . . . . . . 14 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
31 inss2 3867 . . . . . . . . . . . . . 14 (ran 𝑢 ∩ ran (𝐴 × 𝐴)) ⊆ ran (𝐴 × 𝐴)
3230, 31sstri 3645 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝐴 × 𝐴)
33 rnxpid 5602 . . . . . . . . . . . . 13 ran (𝐴 × 𝐴) = 𝐴
3432, 33sseqtri 3670 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴
3534a1i 11 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴)
3627, 35ssind 3870 . . . . . . . . . 10 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎𝐴))
3736adantl 481 . . . . . . . . 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))
38 simpllr 815 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑏 = (𝑎𝐴))
3937, 38sseqtr4d 3675 . . . . . . . 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)
40 imaeq1 5496 . . . . . . . . . 10 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → (𝑣 “ {𝑥}) = ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}))
4140sseq1d 3665 . . . . . . . . 9 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏))
4241rspcev 3340 . . . . . . . 8 (((𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
4322, 39, 42syl2anc 694 . . . . . . 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)
44 simplr 807 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑎 ∈ (unifTop‘𝑈))
45 inss1 3866 . . . . . . . . 9 (𝑎𝐴) ⊆ 𝑎
46 simpllr 815 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑏)
47 simpr 476 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑏 = (𝑎𝐴))
4846, 47eleqtrd 2732 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥 ∈ (𝑎𝐴))
4945, 48sseldi 3634 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑎)
50 elutop 22084 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑎 ∈ (unifTop‘𝑈) ↔ (𝑎𝑋 ∧ ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)))
5150simplbda 653 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) → ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5251r19.21bi 2961 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑥𝑎) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5315, 44, 49, 52syl21anc 1365 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5443, 53r19.29a 3107 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
559adantr 480 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
5654, 55r19.29a 3107 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
5756ralrimiva 2995 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
58 trust 22080 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
59 elutop 22084 . . . . . 6 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
6058, 59syl 17 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
6160biimpar 501 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
621, 14, 57, 61syl12anc 1364 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
6362ex 449 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))))
6463ssrdv 3642 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  {csn 4210   × cxp 5141  ran crn 5144  cima 5146  cfv 5926  (class class class)co 6690  t crest 16128  UnifOncust 22050  unifTopcutop 22081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-rest 16130  df-ust 22051  df-utop 22082
This theorem is referenced by:  restutopopn  22089
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