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Theorem utoptop 21951
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 477 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ (unifTop‘𝑈))
2 utopval 21949 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎})
3 ssrab2 3668 . . . . . . . . 9 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎} ⊆ 𝒫 𝑋
42, 3syl6eqss 3636 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
54adantr 481 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
61, 5sstrd 3594 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ 𝒫 𝑋)
7 sspwuni 4579 . . . . . 6 (𝑥 ⊆ 𝒫 𝑋 𝑥𝑋)
86, 7sylib 208 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥𝑋)
9 simp-4l 805 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑈 ∈ (UnifOn‘𝑋))
10 simp-4r 806 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑥 ⊆ (unifTop‘𝑈))
11 simplr 791 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦𝑥)
1210, 11sseldd 3585 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦 ∈ (unifTop‘𝑈))
13 simpr 477 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑝𝑦)
14 elutop 21950 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → (𝑦 ∈ (unifTop‘𝑈) ↔ (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)))
1514biimpa 501 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
1615simprd 479 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
1716r19.21bi 2927 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
189, 12, 13, 17syl21anc 1322 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
19 r19.41v 3081 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) ↔ (∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥))
20 ssuni 4427 . . . . . . . . . 10 (((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → (𝑣 “ {𝑝}) ⊆ 𝑥)
2120reximi 3005 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2219, 21sylbir 225 . . . . . . . 8 ((∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2318, 11, 22syl2anc 692 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
24 eluni2 4408 . . . . . . . . 9 (𝑝 𝑥 ↔ ∃𝑦𝑥 𝑝𝑦)
2524biimpi 206 . . . . . . . 8 (𝑝 𝑥 → ∃𝑦𝑥 𝑝𝑦)
2625adantl 482 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑦𝑥 𝑝𝑦)
2723, 26r19.29a 3071 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2827ralrimiva 2960 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
29 elutop 21950 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
3029adantr 481 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
318, 28, 30mpbir2and 956 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ∈ (unifTop‘𝑈))
3231ex 450 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
3332alrimiv 1852 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
34 elutop 21950 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ∈ (unifTop‘𝑈) ↔ (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)))
3534biimpa 501 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥))
3635simpld 475 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → 𝑥𝑋)
3736adantrr 752 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → 𝑥𝑋)
38 ssinss1 3821 . . . . 5 (𝑥𝑋 → (𝑥𝑦) ⊆ 𝑋)
3937, 38syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ⊆ 𝑋)
40 simpl 473 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑈 ∈ (UnifOn‘𝑋))
41 simpr31 1149 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑢𝑈)
42 simpr32 1150 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑣𝑈)
43 ustincl 21924 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈𝑣𝑈) → (𝑢𝑣) ∈ 𝑈)
4440, 41, 42, 43syl3anc 1323 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢𝑣) ∈ 𝑈)
45 inss1 3813 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑢
46 imass1 5461 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑢 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝}))
4745, 46ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝})
48 simpr33 1151 . . . . . . . . . . . 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 475 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢 “ {𝑝}) ⊆ 𝑥)
5047, 49syl5ss 3595 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑥)
51 inss2 3814 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑣
52 imass1 5461 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑣 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝}))
5351, 52ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝})
5448simprd 479 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑣 “ {𝑝}) ⊆ 𝑦)
5553, 54syl5ss 3595 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑦)
5650, 55ssind 3817 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦))
57 imaeq1 5422 . . . . . . . . . . 11 (𝑤 = (𝑢𝑣) → (𝑤 “ {𝑝}) = ((𝑢𝑣) “ {𝑝}))
5857sseq1d 3613 . . . . . . . . . 10 (𝑤 = (𝑢𝑣) → ((𝑤 “ {𝑝}) ⊆ (𝑥𝑦) ↔ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)))
5958rspcev 3295 . . . . . . . . 9 (((𝑢𝑣) ∈ 𝑈 ∧ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
6044, 56, 59syl2anc 692 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
61603anassrs 1287 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
62613anassrs 1287 . . . . . 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 789 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑈 ∈ (UnifOn‘𝑋))
64 simplrl 799 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑥 ∈ (unifTop‘𝑈))
65 simpr 477 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝 ∈ (𝑥𝑦))
66 elin 3776 . . . . . . . . . 10 (𝑝 ∈ (𝑥𝑦) ↔ (𝑝𝑥𝑝𝑦))
6765, 66sylib 208 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → (𝑝𝑥𝑝𝑦))
6867simpld 475 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑥)
6935simprd 479 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7069r19.21bi 2927 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) ∧ 𝑝𝑥) → ∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7163, 64, 68, 70syl21anc 1322 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
72 simplrr 800 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑦 ∈ (unifTop‘𝑈))
7367simprd 479 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑦)
7463, 72, 73, 17syl21anc 1322 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
75 reeanv 3097 . . . . . . 7 (∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦) ↔ (∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥 ∧ ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
7671, 74, 75sylanbrc 697 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
7762, 76r19.29vva 3073 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
7877ralrimiva 2960 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
79 elutop 21950 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8079adantr 481 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8139, 78, 80mpbir2and 956 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ∈ (unifTop‘𝑈))
8281ralrimivva 2965 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥𝑦) ∈ (unifTop‘𝑈))
83 fvex 6160 . . 3 (unifTop‘𝑈) ∈ V
84 istopg 20622 . . 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 697 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cin 3555  wss 3556  𝒫 cpw 4132  {csn 4150   cuni 4404  cima 5079  cfv 5849  Topctop 20620  UnifOncust 21916  unifTopcutop 21947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-top 20621  df-ust 21917  df-utop 21948
This theorem is referenced by:  utoptopon  21953  utop2nei  21967  utop3cls  21968  utopreg  21969
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