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Theorem restutopopn 24156
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‘(𝑈t (𝐴 × 𝐴))))

Proof of Theorem restutopopn
Dummy variables 𝑎 𝑏 𝑡 𝑢 𝑤 𝑥 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elutop 24151 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
21simprbda 497 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴𝑋)
3 restutop 24155 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
42, 3syldan 589 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
5 trust 24147 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
62, 5syldan 589 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7 elutop 24151 . . . . . . . 8 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)))
86, 7syl 17 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)))
98simprbda 497 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝐴)
102adantr 479 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
119, 10sstrd 3984 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝑋)
12 simp-9l 791 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑈 ∈ (UnifOn‘𝑋))
13 simplr 767 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑡𝑈)
14 simp-4r 782 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑤𝑈)
15 ustincl 24125 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡𝑈𝑤𝑈) → (𝑡𝑤) ∈ 𝑈)
1612, 13, 14, 15syl3anc 1368 . . . . . . . . . 10 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑡𝑤) ∈ 𝑈)
17 inimass 6155 . . . . . . . . . . 11 ((𝑡𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18 ssrin 4229 . . . . . . . . . . . . . 14 ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
1918adantl 480 . . . . . . . . . . . . 13 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
20 simpllr 774 . . . . . . . . . . . . . . 15 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
2120imaeq1d 6058 . . . . . . . . . . . . . 14 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) = ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}))
229ad5antr 732 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → 𝑏𝐴)
23 simp-5r 784 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → 𝑥𝑏)
2422, 23sseldd 3974 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → 𝑥𝐴)
2524ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑥𝐴)
26 inimasn 6156 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
2726elv 3469 . . . . . . . . . . . . . . . . 17 ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
28 xpimasn 6185 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
2928ineq2d 4207 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
3027, 29eqtrid 2777 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
31 incom 4196 . . . . . . . . . . . . . . . 16 ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
3230, 31eqtrdi 2781 . . . . . . . . . . . . . . 15 (𝑥𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
3325, 32syl 17 . . . . . . . . . . . . . 14 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
3421, 33eqtrd 2765 . . . . . . . . . . . . 13 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
3519, 34sseqtrrd 4015 . . . . . . . . . . . 12 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝑢 “ {𝑥}))
36 simp-5r 784 . . . . . . . . . . . 12 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) ⊆ 𝑏)
3735, 36sstrd 3984 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ 𝑏)
3817, 37sstrid 3985 . . . . . . . . . 10 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏)
39 imaeq1 6054 . . . . . . . . . . . 12 (𝑣 = (𝑡𝑤) → (𝑣 “ {𝑥}) = ((𝑡𝑤) “ {𝑥}))
4039sseq1d 4005 . . . . . . . . . . 11 (𝑣 = (𝑡𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏))
4140rspcev 3603 . . . . . . . . . 10 (((𝑡𝑤) ∈ 𝑈 ∧ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
4216, 38, 41syl2anc 582 . . . . . . . . 9 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
43 simp-4l 781 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)))
4443ad2antrr 724 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)))
451simplbda 498 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4645r19.21bi 3239 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥𝐴) → ∃𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4744, 24, 46syl2anc 582 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → ∃𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4842, 47r19.29a 3152 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) ∧ 𝑤𝑈) ∧ 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
49 simplr 767 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → 𝑢 ∈ (𝑈t (𝐴 × 𝐴)))
50 sqxpexg 7752 . . . . . . . . . . 11 (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
51 elrest 17403 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5250, 51sylan2 591 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5352biimpa 475 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
5443, 49, 53syl2anc 582 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
5548, 54r19.29a 3152 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
568simplbda 498 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5756r19.21bi 3239 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5855, 57r19.29a 3152 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
5958ralrimiva 3136 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
60 elutop 24151 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏𝑋 ∧ ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
6160ad2antrr 724 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏𝑋 ∧ ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
6211, 59, 61mpbir2and 711 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 ∈ (unifTop‘𝑈))
63 dfss2 3959 . . . . . 6 (𝑏𝐴 ↔ (𝑏𝐴) = 𝑏)
649, 63sylib 217 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏𝐴) = 𝑏)
6564eqcomd 2731 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 = (𝑏𝐴))
66 ineq1 4200 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐴) = (𝑏𝐴))
6766rspceeqv 3625 . . . 4 ((𝑏 ∈ (unifTop‘𝑈) ∧ 𝑏 = (𝑏𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
6862, 65, 67syl2anc 582 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
69 fvex 6903 . . . . 5 (unifTop‘𝑈) ∈ V
70 elrest 17403 . . . . 5 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7169, 70mpan 688 . . . 4 (𝐴 ∈ (unifTop‘𝑈) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7271ad2antlr 725 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7368, 72mpbird 256 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴))
744, 73eqelssd 3995 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((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 3051  wrex 3060  Vcvv 3463  cin 3940  wss 3941  {csn 4625   × cxp 5671  cima 5676  cfv 6543  (class class class)co 7413  t crest 17396  UnifOncust 24117  unifTopcutop 24148
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
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 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-rest 17398  df-ust 24118  df-utop 24149
This theorem is referenced by:  ressusp  24182
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