MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restutopopn Structured version   Visualization version   GIF version

Theorem restutopopn 24117
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 24112 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)))
21simprbda 498 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → 𝐴𝑋)
3 restutop 24116 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
42, 3syldan 590 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
5 trust 24108 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
62, 5syldan 590 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
7 elutop 24112 . . . . . . . 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 498 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏𝐴)
102adantr 480 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
119, 10sstrd 3988 . . . . 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 24086 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑡𝑈𝑤𝑈) → (𝑡𝑤) ∈ 𝑈)
1612, 13, 14, 15syl3anc 1369 . . . . . . . . . 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 6153 . . . . . . . . . . 11 ((𝑡𝑤) “ {𝑥}) ⊆ ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥}))
18 ssrin 4229 . . . . . . . . . . . . . 14 ((𝑡 “ {𝑥}) ⊆ 𝐴 → ((𝑡 “ {𝑥}) ∩ (𝑤 “ {𝑥})) ⊆ (𝐴 ∩ (𝑤 “ {𝑥})))
1918adantl 481 . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . 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 3979 . . . . . . . . . . . . . . . 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 6154 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})))
2726elv 3475 . . . . . . . . . . . . . . . . 17 ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥}))
28 xpimasn 6183 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝐴 × 𝐴) “ {𝑥}) = 𝐴)
2928ineq2d 4208 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑤 “ {𝑥}) ∩ ((𝐴 × 𝐴) “ {𝑥})) = ((𝑤 “ {𝑥}) ∩ 𝐴))
3027, 29eqtrid 2779 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑤 ∩ (𝐴 × 𝐴)) “ {𝑥}) = ((𝑤 “ {𝑥}) ∩ 𝐴))
31 incom 4197 . . . . . . . . . . . . . . . 16 ((𝑤 “ {𝑥}) ∩ 𝐴) = (𝐴 ∩ (𝑤 “ {𝑥}))
3230, 31eqtrdi 2783 . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . 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 4019 . . . . . . . . . . . 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 3988 . . . . . . . . . . 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 3989 . . . . . . . . . 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 6052 . . . . . . . . . . . 12 (𝑣 = (𝑡𝑤) → (𝑣 “ {𝑥}) = ((𝑡𝑤) “ {𝑥}))
4039sseq1d 4009 . . . . . . . . . . 11 (𝑣 = (𝑡𝑤) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏))
4140rspcev 3607 . . . . . . . . . 10 (((𝑡𝑤) ∈ 𝑈 ∧ ((𝑡𝑤) “ {𝑥}) ⊆ 𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
4216, 38, 41syl2anc 583 . . . . . . . . 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 499 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ∀𝑥𝐴𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4645r19.21bi 3243 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑥𝐴) → ∃𝑡𝑈 (𝑡 “ {𝑥}) ⊆ 𝐴)
4744, 24, 46syl2anc 583 . . . . . . . . 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 3157 . . . . . . . 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 7749 . . . . . . . . . . 11 (𝐴 ∈ (unifTop‘𝑈) → (𝐴 × 𝐴) ∈ V)
51 elrest 17394 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5250, 51sylan2 592 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → (𝑢 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴))))
5352biimpa 476 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑢 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑤𝑈 𝑢 = (𝑤 ∩ (𝐴 × 𝐴)))
5443, 49, 53syl2anc 583 . . . . . . . 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 3157 . . . . . . 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 499 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5756r19.21bi 3243 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑢 ∈ (𝑈t (𝐴 × 𝐴))(𝑢 “ {𝑥}) ⊆ 𝑏)
5855, 57r19.29a 3157 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑥𝑏) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
5958ralrimiva 3141 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑥𝑏𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑏)
60 elutop 24112 . . . . . 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 3961 . . . . . 6 (𝑏𝐴 ↔ (𝑏𝐴) = 𝑏)
649, 63sylib 217 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → (𝑏𝐴) = 𝑏)
6564eqcomd 2733 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → 𝑏 = (𝑏𝐴))
66 ineq1 4201 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐴) = (𝑏𝐴))
6766rspceeqv 3629 . . . 4 ((𝑏 ∈ (unifTop‘𝑈) ∧ 𝑏 = (𝑏𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
6862, 65, 67syl2anc 583 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) ∧ 𝑏 ∈ (unifTop‘(𝑈t (𝐴 × 𝐴)))) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎𝐴))
69 fvex 6904 . . . . 5 (unifTop‘𝑈) ∈ V
70 elrest 17394 . . . . 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 3999 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 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  Vcvv 3469  cin 3943  wss 3944  {csn 4624   × cxp 5670  cima 5675  cfv 6542  (class class class)co 7414  t crest 17387  UnifOncust 24078  unifTopcutop 24109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-rest 17389  df-ust 24079  df-utop 24110
This theorem is referenced by:  ressusp  24143
  Copyright terms: Public domain W3C validator