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Theorem restutop 22320
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 474 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋))
2 fvexd 6390 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (unifTop‘𝑈) ∈ V)
3 elfvex 6409 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
43adantr 472 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
5 simpr 477 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
64, 5ssexd 4966 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
7 elrest 16356 . . . . . . 7 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ V) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
82, 6, 7syl2anc 579 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
98biimpa 468 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
10 inss2 3993 . . . . . . 7 (𝑎𝐴) ⊆ 𝐴
11 sseq1 3786 . . . . . . 7 (𝑏 = (𝑎𝐴) → (𝑏𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
1210, 11mpbiri 249 . . . . . 6 (𝑏 = (𝑎𝐴) → 𝑏𝐴)
1312rexlimivw 3176 . . . . 5 (∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴) → 𝑏𝐴)
149, 13syl 17 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏𝐴)
15 simp-5l 805 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑈 ∈ (UnifOn‘𝑋))
1615ad2antrr 717 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
176ad6antr 732 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝐴 ∈ V)
18 xpexg 7158 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
1917, 17, 18syl2anc 579 . . . . . . . . 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 785 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑢𝑈)
21 elrestr 16357 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑢𝑈) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2216, 19, 20, 21syl3anc 1490 . . . . . . . 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 3992 . . . . . . . . . . . . 13 (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢
24 imass1 5682 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}))
2523, 24ax-mp 5 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥})
26 sstr 3769 . . . . . . . . . . . 12 ((((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
2725, 26mpan 681 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
28 imassrn 5659 . . . . . . . . . . . . . . 15 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝑢 ∩ (𝐴 × 𝐴))
29 rnin 5725 . . . . . . . . . . . . . . 15 ran (𝑢 ∩ (𝐴 × 𝐴)) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
3028, 29sstri 3770 . . . . . . . . . . . . . 14 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
31 inss2 3993 . . . . . . . . . . . . . 14 (ran 𝑢 ∩ ran (𝐴 × 𝐴)) ⊆ ran (𝐴 × 𝐴)
3230, 31sstri 3770 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝐴 × 𝐴)
33 rnxpid 5750 . . . . . . . . . . . . 13 ran (𝐴 × 𝐴) = 𝐴
3432, 33sseqtri 3797 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴
3534a1i 11 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴)
3627, 35ssind 3996 . . . . . . . . . 10 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎𝐴))
3736adantl 473 . . . . . . . . 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 793 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑏 = (𝑎𝐴))
3937, 38sseqtr4d 3802 . . . . . . . 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 5643 . . . . . . . . . 10 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → (𝑣 “ {𝑥}) = ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}))
4140sseq1d 3792 . . . . . . . . 9 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏))
4241rspcev 3461 . . . . . . . 8 (((𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
4322, 39, 42syl2anc 579 . . . . . . 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 785 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑎 ∈ (unifTop‘𝑈))
45 inss1 3992 . . . . . . . . 9 (𝑎𝐴) ⊆ 𝑎
46 simpllr 793 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑏)
47 simpr 477 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑏 = (𝑎𝐴))
4846, 47eleqtrd 2846 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥 ∈ (𝑎𝐴))
4945, 48sseldi 3759 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑎)
50 elutop 22316 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑎 ∈ (unifTop‘𝑈) ↔ (𝑎𝑋 ∧ ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)))
5150simplbda 493 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) → ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5251r19.21bi 3079 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑥𝑎) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5315, 44, 49, 52syl21anc 866 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5443, 53r19.29a 3225 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
559adantr 472 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
5654, 55r19.29a 3225 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
5756ralrimiva 3113 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
58 trust 22312 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
59 elutop 22316 . . . . . 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 469 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
621, 14, 57, 61syl12anc 865 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
6362ex 401 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))))
6463ssrdv 3767 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cin 3731  wss 3732  {csn 4334   × cxp 5275  ran crn 5278  cima 5280  cfv 6068  (class class class)co 6842  t crest 16349  UnifOncust 22282  unifTopcutop 22313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-rest 16351  df-ust 22283  df-utop 22314
This theorem is referenced by:  restutopopn  22321
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