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Theorem utoptop 23609
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 486 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ (unifTop‘𝑈))
2 utopval 23607 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎})
3 ssrab2 4041 . . . . . . . . 9 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎} ⊆ 𝒫 𝑋
42, 3eqsstrdi 4002 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
54adantr 482 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
61, 5sstrd 3958 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ 𝒫 𝑋)
7 sspwuni 5064 . . . . . 6 (𝑥 ⊆ 𝒫 𝑋 𝑥𝑋)
86, 7sylib 217 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥𝑋)
9 simp-4l 782 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑈 ∈ (UnifOn‘𝑋))
10 simp-4r 783 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑥 ⊆ (unifTop‘𝑈))
11 simplr 768 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦𝑥)
1210, 11sseldd 3949 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦 ∈ (unifTop‘𝑈))
13 simpr 486 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑝𝑦)
14 elutop 23608 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → (𝑦 ∈ (unifTop‘𝑈) ↔ (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)))
1514biimpa 478 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
1615simprd 497 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
1716r19.21bi 3233 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
189, 12, 13, 17syl21anc 837 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
19 r19.41v 3182 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) ↔ (∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥))
20 ssuni 4897 . . . . . . . . . 10 (((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → (𝑣 “ {𝑝}) ⊆ 𝑥)
2120reximi 3084 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2219, 21sylbir 234 . . . . . . . 8 ((∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2318, 11, 22syl2anc 585 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
24 eluni2 4873 . . . . . . . . 9 (𝑝 𝑥 ↔ ∃𝑦𝑥 𝑝𝑦)
2524biimpi 215 . . . . . . . 8 (𝑝 𝑥 → ∃𝑦𝑥 𝑝𝑦)
2625adantl 483 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑦𝑥 𝑝𝑦)
2723, 26r19.29a 3156 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2827ralrimiva 3140 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
29 elutop 23608 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
3029adantr 482 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
318, 28, 30mpbir2and 712 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ∈ (unifTop‘𝑈))
3231ex 414 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
3332alrimiv 1931 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
34 elutop 23608 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ∈ (unifTop‘𝑈) ↔ (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)))
3534biimpa 478 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥))
3635simpld 496 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → 𝑥𝑋)
3736adantrr 716 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → 𝑥𝑋)
38 ssinss1 4201 . . . . 5 (𝑥𝑋 → (𝑥𝑦) ⊆ 𝑋)
3937, 38syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ⊆ 𝑋)
40 simpl 484 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑈 ∈ (UnifOn‘𝑋))
41 simpr31 1264 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑢𝑈)
42 simpr32 1265 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑣𝑈)
43 ustincl 23582 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈𝑣𝑈) → (𝑢𝑣) ∈ 𝑈)
4440, 41, 42, 43syl3anc 1372 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢𝑣) ∈ 𝑈)
45 inss1 4192 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑢
46 imass1 6057 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑢 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝}))
4745, 46ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝})
48 simpr33 1266 . . . . . . . . . . . 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 496 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢 “ {𝑝}) ⊆ 𝑥)
5047, 49sstrid 3959 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑥)
51 inss2 4193 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑣
52 imass1 6057 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑣 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝}))
5351, 52ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝})
5448simprd 497 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑣 “ {𝑝}) ⊆ 𝑦)
5553, 54sstrid 3959 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑦)
5650, 55ssind 4196 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦))
57 imaeq1 6012 . . . . . . . . . . 11 (𝑤 = (𝑢𝑣) → (𝑤 “ {𝑝}) = ((𝑢𝑣) “ {𝑝}))
5857sseq1d 3979 . . . . . . . . . 10 (𝑤 = (𝑢𝑣) → ((𝑤 “ {𝑝}) ⊆ (𝑥𝑦) ↔ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)))
5958rspcev 3583 . . . . . . . . 9 (((𝑢𝑣) ∈ 𝑈 ∧ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
6044, 56, 59syl2anc 585 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
61603anassrs 1361 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
62613anassrs 1361 . . . . . 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 766 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑈 ∈ (UnifOn‘𝑋))
64 simplrl 776 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑥 ∈ (unifTop‘𝑈))
65 simpr 486 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝 ∈ (𝑥𝑦))
66 elin 3930 . . . . . . . . . 10 (𝑝 ∈ (𝑥𝑦) ↔ (𝑝𝑥𝑝𝑦))
6765, 66sylib 217 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → (𝑝𝑥𝑝𝑦))
6867simpld 496 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑥)
6935simprd 497 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7069r19.21bi 3233 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) ∧ 𝑝𝑥) → ∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7163, 64, 68, 70syl21anc 837 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
72 simplrr 777 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑦 ∈ (unifTop‘𝑈))
7367simprd 497 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑦)
7463, 72, 73, 17syl21anc 837 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
75 reeanv 3216 . . . . . . 7 (∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦) ↔ (∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥 ∧ ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
7671, 74, 75sylanbrc 584 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
7762, 76r19.29vva 3204 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
7877ralrimiva 3140 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
79 elutop 23608 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8079adantr 482 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8139, 78, 80mpbir2and 712 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ∈ (unifTop‘𝑈))
8281ralrimivva 3194 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥𝑦) ∈ (unifTop‘𝑈))
83 fvex 6859 . . 3 (unifTop‘𝑈) ∈ V
84 istopg 22267 . . 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 584 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  Vcvv 3447  cin 3913  wss 3914  𝒫 cpw 4564  {csn 4590   cuni 4869  cima 5640  cfv 6500  Topctop 22265  UnifOncust 23574  unifTopcutop 23605
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508  df-top 22266  df-ust 23575  df-utop 23606
This theorem is referenced by:  utoptopon  23611  utop2nei  23625  utop3cls  23626  utopreg  23627
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