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Theorem restutopopn 22089
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 22084 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
21simprbda 652 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴𝑋)
3 restutop 22088 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
42, 3syldan 486 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
5 trust 22080 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
62, 5syldan 486 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7 elutop 22084 . . . . . . . . . 10 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)))
86, 7syl 17 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) ↔ (𝑏𝐴 ∧ ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)))
98simprbda 652 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝐴)
102adantr 480 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
119, 10sstrd 3646 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝑋)
12 simp-9l 833 . . . . . . . . . . . . 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 ∈ (UnifOn‘𝑋))
13 simplr 807 . . . . . . . . . . . . 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) → 𝑡𝑈)
14 simp-4r 824 . . . . . . . . . . . . 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) → 𝑤𝑈)
15 ustincl 22058 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡𝑈𝑤𝑈) → (𝑡𝑤) ∈ 𝑈)
1612, 13, 14, 15syl3anc 1366 . . . . . . . . . . . 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) → (𝑡𝑤) ∈ 𝑈)
17 inimass 5584 . . . . . . . . . . . . 13 ((𝑡𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18 ssrin 3871 . . . . . . . . . . . . . . . 16 ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
1918adantl 481 . . . . . . . . . . . . . . 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) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
20 simpllr 815 . . . . . . . . . . . . . . . . 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))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
2120imaeq1d 5500 . . . . . . . . . . . . . . . 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))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → (𝑢 “ {𝑥}) = ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}))
229ad5antr 773 . . . . . . . . . . . . . . . . . . 19 ((((((((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 826 . . . . . . . . . . . . . . . . . . 19 ((((((((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 3637 . . . . . . . . . . . . . . . . . 18 ((((((((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 762 . . . . . . . . . . . . . . . . 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))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → 𝑥𝐴)
26 vex 3234 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
27 inimasn 5585 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
2826, 27ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
29 xpimasn 5614 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
3029ineq2d 3847 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
3128, 30syl5eq 2697 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
32 incom 3838 . . . . . . . . . . . . . . . . . 18 ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
3331, 32syl6eq 2701 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
3425, 33syl 17 . . . . . . . . . . . . . . . 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))) ∧ 𝑡𝑈) ∧ (𝑡 “ {𝑥}) ⊆ 𝐴) → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
3521, 34eqtrd 2685 . . . . . . . . . . . . . . 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 “ {𝑥}) = (𝐴 ∩ (𝑤 “ {𝑥})))
3619, 35sseqtr4d 3675 . . . . . . . . . . . . . 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) → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝑢 “ {𝑥}))
37 simp-5r 826 . . . . . . . . . . . . . 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}) ⊆ 𝑏)
3836, 37sstrd 3646 . . . . . . . . . . . . 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})) ⊆ 𝑏)
3917, 38syl5ss 3647 . . . . . . . . . . . 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) → ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏)
40 imaeq1 5496 . . . . . . . . . . . . . 14 (𝑣 = (𝑡𝑤) → (𝑣 “ {𝑥}) = ((𝑡𝑤) “ {𝑥}))
4140sseq1d 3665 . . . . . . . . . . . . 13 (𝑣 = (𝑡𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏))
4241rspcev 3340 . . . . . . . . . . . 12 (((𝑡𝑤) ∈ 𝑈 ∧ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
4316, 39, 42syl2anc 694 . . . . . . . . . . 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) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
44 simp-4l 823 . . . . . . . . . . . . 13 ((((((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)))
4544ad2antrr 762 . . . . . . . . . . . 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))) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)))
461simplbda 653 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4746r19.21bi 2961 . . . . . . . . . . . 12 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥𝐴) → ∃𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4845, 24, 47syl2anc 694 . . . . . . . . . . 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)
4943, 48r19.29a 3107 . . . . . . . . . 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))) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
50 simplr 807 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → 𝑢 ∈ (𝑈t (𝐴 × 𝐴)))
51 sqxpexg 7005 . . . . . . . . . . . . 13 (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
52 elrest 16135 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5351, 52sylan2 490 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5453biimpa 500 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
5544, 50, 54syl2anc 694 . . . . . . . . . 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)))
5649, 55r19.29a 3107 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) ∧ (𝑢 “ {𝑥}) ⊆ 𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
578simplbda 653 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5857r19.21bi 2961 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5956, 58r19.29a 3107 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
6059ralrimiva 2995 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
61 elutop 22084 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏𝑋 ∧ ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
6261ad2antrr 762 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏 ∈ (unifTop‘𝑈) ↔ (𝑏𝑋 ∧ ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)))
6311, 60, 62mpbir2and 977 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 ∈ (unifTop‘𝑈))
64 df-ss 3621 . . . . . . . 8 (𝑏𝐴 ↔ (𝑏𝐴) = 𝑏)
659, 64sylib 208 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏𝐴) = 𝑏)
6665eqcomd 2657 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 = (𝑏𝐴))
67 ineq1 3840 . . . . . . . 8 (𝑎 = 𝑏 → (𝑎𝐴) = (𝑏𝐴))
6867eqeq2d 2661 . . . . . . 7 (𝑎 = 𝑏 → (𝑏 = (𝑎𝐴) ↔ 𝑏 = (𝑏𝐴)))
6968rspcev 3340 . . . . . 6 ((𝑏 ∈ (unifTop‘𝑈) ∧ 𝑏 = (𝑏𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
7063, 66, 69syl2anc 694 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
71 fvex 6239 . . . . . . 7 (unifTop‘𝑈) ∈ V
72 elrest 16135 . . . . . . 7 (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7371, 72mpan 706 . . . . . 6 (𝐴 ∈ (unifTop‘𝑈) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7473ad2antlr 763 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴)))
7570, 74mpbird 247 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴))
7675ex 449 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴))) → 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)))
7776ssrdv 3642 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (unifTop‘(𝑈t (𝐴 × 𝐴))) ⊆ ((unifTop‘𝑈) ↾t 𝐴))
784, 77eqssd 3653 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  {csn 4210   × cxp 5141  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:  ressusp  22116
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