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Theorem restutop 21851
 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 fvex 6113 . . . . . . . 8 (unifTop‘𝑈) ∈ V
32a1i 11 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (unifTop‘𝑈) ∈ V)
4 elfvex 6131 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
54adantr 480 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
6 simpr 476 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
75, 6ssexd 4733 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
8 elrest 15911 . . . . . . 7 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ V) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
93, 7, 8syl2anc 691 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
109biimpa 500 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
11 inss2 3796 . . . . . . 7 (𝑎𝐴) ⊆ 𝐴
12 sseq1 3589 . . . . . . 7 (𝑏 = (𝑎𝐴) → (𝑏𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
1311, 12mpbiri 247 . . . . . 6 (𝑏 = (𝑎𝐴) → 𝑏𝐴)
1413rexlimivw 3011 . . . . 5 (∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴) → 𝑏𝐴)
1510, 14syl 17 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏𝐴)
16 simp-5l 804 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑈 ∈ (UnifOn‘𝑋))
1716ad2antrr 758 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
187ad6antr 768 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝐴 ∈ V)
19 xpexg 6858 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
2018, 18, 19syl2anc 691 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝐴 × 𝐴) ∈ V)
21 simplr 788 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑢𝑈)
22 elrestr 15912 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑢𝑈) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2317, 20, 21, 22syl3anc 1318 . . . . . . . 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)))
24 inss1 3795 . . . . . . . . . . . . 13 (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢
25 imass1 5419 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥})
27 sstr 3576 . . . . . . . . . . . 12 ((((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
2826, 27mpan 702 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
29 imassrn 5396 . . . . . . . . . . . . . . 15 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝑢 ∩ (𝐴 × 𝐴))
30 rnin 5461 . . . . . . . . . . . . . . 15 ran (𝑢 ∩ (𝐴 × 𝐴)) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
3129, 30sstri 3577 . . . . . . . . . . . . . 14 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
32 inss2 3796 . . . . . . . . . . . . . 14 (ran 𝑢 ∩ ran (𝐴 × 𝐴)) ⊆ ran (𝐴 × 𝐴)
3331, 32sstri 3577 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝐴 × 𝐴)
34 rnxpid 5486 . . . . . . . . . . . . 13 ran (𝐴 × 𝐴) = 𝐴
3533, 34sseqtri 3600 . . . . . . . . . . . 12 ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴
3635a1i 11 . . . . . . . . . . 11 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴)
3728, 36ssind 3799 . . . . . . . . . 10 ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎𝐴))
3837adantl 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))
39 simpllr 795 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) ∧ 𝑢𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑏 = (𝑎𝐴))
4038, 39sseqtr4d 3605 . . . . . . . 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)
41 imaeq1 5380 . . . . . . . . . 10 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → (𝑣 “ {𝑥}) = ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}))
4241sseq1d 3595 . . . . . . . . 9 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏))
4342rspcev 3282 . . . . . . . 8 (((𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
4423, 40, 43syl2anc 691 . . . . . . 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)
45 simplr 788 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑎 ∈ (unifTop‘𝑈))
46 inss1 3795 . . . . . . . . 9 (𝑎𝐴) ⊆ 𝑎
47 simpllr 795 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑏)
48 simpr 476 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑏 = (𝑎𝐴))
4947, 48eleqtrd 2690 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥 ∈ (𝑎𝐴))
5046, 49sseldi 3566 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → 𝑥𝑎)
51 elutop 21847 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑎 ∈ (unifTop‘𝑈) ↔ (𝑎𝑋 ∧ ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)))
5251simplbda 652 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) → ∀𝑥𝑎𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5352r19.21bi 2916 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑥𝑎) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5416, 45, 50, 53syl21anc 1317 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑢𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
5544, 54r19.29a 3060 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎𝐴)) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
5610adantr 480 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
5755, 56r19.29a 3060 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥𝑏) → ∃𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
5857ralrimiva 2949 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
59 trust 21843 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
60 elutop 21847 . . . . . 6 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
6159, 60syl 17 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
6261biimpar 501 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ (𝑏𝐴 ∧ ∀𝑥𝑏𝑣 ∈ (𝑈t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
631, 15, 58, 62syl12anc 1316 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))))
6463ex 449 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) → 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))))
6564ssrdv 3574 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
 Colors of variables: wff setvar class 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  ran crn 5039   “ 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:  restutopopn  21852
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