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

Theorem utoptop 21848
Description: The topology induced by a uniform structure 𝑈 is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
Assertion
Ref Expression
utoptop (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)

Proof of Theorem utoptop
Dummy variables 𝑝 𝑎 𝑢 𝑣 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ (unifTop‘𝑈))
2 utopval 21846 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎})
3 ssrab2 3650 . . . . . . . . 9 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎} ⊆ 𝒫 𝑋
42, 3syl6eqss 3618 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
54adantr 480 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
61, 5sstrd 3578 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ 𝒫 𝑋)
7 sspwuni 4547 . . . . . 6 (𝑥 ⊆ 𝒫 𝑋 𝑥𝑋)
86, 7sylib 207 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥𝑋)
9 simp-4l 802 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑈 ∈ (UnifOn‘𝑋))
10 simp-4r 803 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑥 ⊆ (unifTop‘𝑈))
11 simplr 788 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦𝑥)
1210, 11sseldd 3569 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦 ∈ (unifTop‘𝑈))
13 simpr 476 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑝𝑦)
14 elutop 21847 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → (𝑦 ∈ (unifTop‘𝑈) ↔ (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)))
1514biimpa 500 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
1615simprd 478 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
1716r19.21bi 2916 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
189, 12, 13, 17syl21anc 1317 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
19 r19.41v 3070 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) ↔ (∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥))
20 ssuni 4395 . . . . . . . . . 10 (((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → (𝑣 “ {𝑝}) ⊆ 𝑥)
2120reximi 2994 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2219, 21sylbir 224 . . . . . . . 8 ((∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2318, 11, 22syl2anc 691 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
24 eluni2 4376 . . . . . . . . 9 (𝑝 𝑥 ↔ ∃𝑦𝑥 𝑝𝑦)
2524biimpi 205 . . . . . . . 8 (𝑝 𝑥 → ∃𝑦𝑥 𝑝𝑦)
2625adantl 481 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑦𝑥 𝑝𝑦)
2723, 26r19.29a 3060 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2827ralrimiva 2949 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
29 elutop 21847 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
3029adantr 480 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
318, 28, 30mpbir2and 959 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ∈ (unifTop‘𝑈))
3231ex 449 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
3332alrimiv 1842 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
34 elutop 21847 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ∈ (unifTop‘𝑈) ↔ (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)))
3534biimpa 500 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥))
3635simpld 474 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → 𝑥𝑋)
3736adantrr 749 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → 𝑥𝑋)
38 ssinss1 3803 . . . . 5 (𝑥𝑋 → (𝑥𝑦) ⊆ 𝑋)
3937, 38syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ⊆ 𝑋)
40 simpl 472 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑈 ∈ (UnifOn‘𝑋))
41 simpr31 1144 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑢𝑈)
42 simpr32 1145 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑣𝑈)
43 ustincl 21821 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈𝑣𝑈) → (𝑢𝑣) ∈ 𝑈)
4440, 41, 42, 43syl3anc 1318 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢𝑣) ∈ 𝑈)
45 inss1 3795 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑢
46 imass1 5419 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑢 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝}))
4745, 46ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝})
48 simpr33 1146 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
4948simpld 474 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢 “ {𝑝}) ⊆ 𝑥)
5047, 49syl5ss 3579 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑥)
51 inss2 3796 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑣
52 imass1 5419 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑣 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝}))
5351, 52ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝})
5448simprd 478 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑣 “ {𝑝}) ⊆ 𝑦)
5553, 54syl5ss 3579 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑦)
5650, 55ssind 3799 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦))
57 imaeq1 5380 . . . . . . . . . . 11 (𝑤 = (𝑢𝑣) → (𝑤 “ {𝑝}) = ((𝑢𝑣) “ {𝑝}))
5857sseq1d 3595 . . . . . . . . . 10 (𝑤 = (𝑢𝑣) → ((𝑤 “ {𝑝}) ⊆ (𝑥𝑦) ↔ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)))
5958rspcev 3282 . . . . . . . . 9 (((𝑢𝑣) ∈ 𝑈 ∧ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
6044, 56, 59syl2anc 691 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
61603anassrs 1282 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
62613anassrs 1282 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) ∧ 𝑢𝑈) ∧ 𝑣𝑈) ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
63 simpll 786 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑈 ∈ (UnifOn‘𝑋))
64 simplrl 796 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑥 ∈ (unifTop‘𝑈))
65 simpr 476 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝 ∈ (𝑥𝑦))
66 elin 3758 . . . . . . . . . 10 (𝑝 ∈ (𝑥𝑦) ↔ (𝑝𝑥𝑝𝑦))
6765, 66sylib 207 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → (𝑝𝑥𝑝𝑦))
6867simpld 474 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑥)
6935simprd 478 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7069r19.21bi 2916 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) ∧ 𝑝𝑥) → ∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7163, 64, 68, 70syl21anc 1317 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
72 simplrr 797 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑦 ∈ (unifTop‘𝑈))
7367simprd 478 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑦)
7463, 72, 73, 17syl21anc 1317 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
75 reeanv 3086 . . . . . . 7 (∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦) ↔ (∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥 ∧ ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
7671, 74, 75sylanbrc 695 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
7762, 76r19.29vva 3062 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
7877ralrimiva 2949 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
79 elutop 21847 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8079adantr 480 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8139, 78, 80mpbir2and 959 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ∈ (unifTop‘𝑈))
8281ralrimivva 2954 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥𝑦) ∈ (unifTop‘𝑈))
83 fvex 6113 . . 3 (unifTop‘𝑈) ∈ V
84 istopg 20525 . . 3 ((unifTop‘𝑈) ∈ V → ((unifTop‘𝑈) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)) ∧ ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥𝑦) ∈ (unifTop‘𝑈))))
8583, 84ax-mp 5 . 2 ((unifTop‘𝑈) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)) ∧ ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥𝑦) ∈ (unifTop‘𝑈)))
8633, 82, 85sylanbrc 695 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108  {csn 4125   cuni 4372  cima 5041  cfv 5804  Topctop 20517  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-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-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-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-fv 5812  df-top 20521  df-ust 21814  df-utop 21845
This theorem is referenced by:  utoptopon  21850  utop2nei  21864  utop3cls  21865  utopreg  21866
  Copyright terms: Public domain W3C validator