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Theorem restutopopn 23606
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 23601 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
21simprbda 500 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴𝑋)
3 restutop 23605 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
42, 3syldan 592 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
5 trust 23597 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
62, 5syldan 592 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7 elutop 23601 . . . . . . . 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 500 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝐴)
102adantr 482 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
119, 10sstrd 3959 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝑋)
12 simp-9l 792 . . . . . . . . . . 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 768 . . . . . . . . . . 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 783 . . . . . . . . . . 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 23575 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡𝑈𝑤𝑈) → (𝑡𝑤) ∈ 𝑈)
1612, 13, 14, 15syl3anc 1372 . . . . . . . . . 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 6112 . . . . . . . . . . 11 ((𝑡𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18 ssrin 4198 . . . . . . . . . . . . . 14 ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
1918adantl 483 . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . . 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 3950 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . 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 6113 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
2726elv 3454 . . . . . . . . . . . . . . . . 17 ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
28 xpimasn 6142 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
2928ineq2d 4177 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
3027, 29eqtrid 2789 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
31 incom 4166 . . . . . . . . . . . . . . . 16 ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
3230, 31eqtrdi 2793 . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 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 3990 . . . . . . . . . . . 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 785 . . . . . . . . . . . 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 3959 . . . . . . . . . . 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 3960 . . . . . . . . . 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 6013 . . . . . . . . . . . 12 (𝑣 = (𝑡𝑤) → (𝑣 “ {𝑥}) = ((𝑡𝑤) “ {𝑥}))
4039sseq1d 3980 . . . . . . . . . . 11 (𝑣 = (𝑡𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏))
4140rspcev 3584 . . . . . . . . . 10 (((𝑡𝑤) ∈ 𝑈 ∧ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
4216, 38, 41syl2anc 585 . . . . . . . . 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 782 . . . . . . . . . . 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 725 . . . . . . . . . 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 501 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4645r19.21bi 3237 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥𝐴) → ∃𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4744, 24, 46syl2anc 585 . . . . . . . . 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 3160 . . . . . . . 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 768 . . . . . . . . 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 7694 . . . . . . . . . . 11 (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
51 elrest 17316 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5250, 51sylan2 594 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5352biimpa 478 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
5443, 49, 53syl2anc 585 . . . . . . . 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 3160 . . . . . . 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 501 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5756r19.21bi 3237 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5855, 57r19.29a 3160 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
5958ralrimiva 3144 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
60 elutop 23601 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏𝑋 ∧ ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
6160ad2antrr 725 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏𝑋 ∧ ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
6211, 59, 61mpbir2and 712 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 ∈ (unifTop‘𝑈))
63 df-ss 3932 . . . . . 6 (𝑏𝐴 ↔ (𝑏𝐴) = 𝑏)
649, 63sylib 217 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏𝐴) = 𝑏)
6564eqcomd 2743 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 = (𝑏𝐴))
66 ineq1 4170 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐴) = (𝑏𝐴))
6766rspceeqv 3600 . . . 4 ((𝑏 ∈ (unifTop‘𝑈) ∧ 𝑏 = (𝑏𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
6862, 65, 67syl2anc 585 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
69 fvex 6860 . . . . 5 (unifTop‘𝑈) ∈ V
70 elrest 17316 . . . . 5 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7169, 70mpan 689 . . . 4 (𝐴 ∈ (unifTop‘𝑈) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7271ad2antlr 726 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7368, 72mpbird 257 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴))
744, 73eqelssd 3970 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 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  cin 3914  wss 3915  {csn 4591   × cxp 5636  cima 5641  cfv 6501  (class class class)co 7362  t crest 17309  UnifOncust 23567  unifTopcutop 23598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-rest 17311  df-ust 23568  df-utop 23599
This theorem is referenced by:  ressusp  23632
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