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Theorem utoptop 22979
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 488 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ (unifTop‘𝑈))
2 utopval 22977 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎})
3 ssrab2 3967 . . . . . . . . 9 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎} ⊆ 𝒫 𝑋
42, 3eqsstrdi 3929 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
54adantr 484 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
61, 5sstrd 3885 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ 𝒫 𝑋)
7 sspwuni 4982 . . . . . 6 (𝑥 ⊆ 𝒫 𝑋 𝑥𝑋)
86, 7sylib 221 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥𝑋)
9 simp-4l 783 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑈 ∈ (UnifOn‘𝑋))
10 simp-4r 784 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑥 ⊆ (unifTop‘𝑈))
11 simplr 769 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦𝑥)
1210, 11sseldd 3876 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑦 ∈ (unifTop‘𝑈))
13 simpr 488 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → 𝑝𝑦)
14 elutop 22978 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → (𝑦 ∈ (unifTop‘𝑈) ↔ (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)))
1514biimpa 480 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → (𝑦𝑋 ∧ ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
1615simprd 499 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → ∀𝑝𝑦𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
1716r19.21bi 3120 . . . . . . . . 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 3250 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) ↔ (∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥))
20 ssuni 4820 . . . . . . . . . 10 (((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → (𝑣 “ {𝑝}) ⊆ 𝑥)
2120reximi 3156 . . . . . . . . 9 (∃𝑣𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2219, 21sylbir 238 . . . . . . . 8 ((∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦𝑦𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2318, 11, 22syl2anc 587 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) ∧ 𝑦𝑥) ∧ 𝑝𝑦) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
24 eluni2 4797 . . . . . . . . 9 (𝑝 𝑥 ↔ ∃𝑦𝑥 𝑝𝑦)
2524biimpi 219 . . . . . . . 8 (𝑝 𝑥 → ∃𝑦𝑥 𝑝𝑦)
2625adantl 485 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑦𝑥 𝑝𝑦)
2723, 26r19.29a 3198 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 𝑥) → ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
2827ralrimiva 3096 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)
29 elutop 22978 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
3029adantr 484 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ( 𝑥 ∈ (unifTop‘𝑈) ↔ ( 𝑥𝑋 ∧ ∀𝑝 𝑥𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑥)))
318, 28, 30mpbir2and 713 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ∈ (unifTop‘𝑈))
3231ex 416 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
3332alrimiv 1933 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → 𝑥 ∈ (unifTop‘𝑈)))
34 elutop 22978 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ∈ (unifTop‘𝑈) ↔ (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)))
3534biimpa 480 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → (𝑥𝑋 ∧ ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥))
3635simpld 498 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → 𝑥𝑋)
3736adantrr 717 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → 𝑥𝑋)
38 ssinss1 4126 . . . . 5 (𝑥𝑋 → (𝑥𝑦) ⊆ 𝑋)
3937, 38syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ⊆ 𝑋)
40 simpl 486 . . . . . . . . . 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 22952 . . . . . . . . . 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 4117 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑢
46 imass1 5932 . . . . . . . . . . . 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 498 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢 “ {𝑝}) ⊆ 𝑥)
5047, 49sstrid 3886 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑥)
51 inss2 4118 . . . . . . . . . . . 12 (𝑢𝑣) ⊆ 𝑣
52 imass1 5932 . . . . . . . . . . . 12 ((𝑢𝑣) ⊆ 𝑣 → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝}))
5351, 52ax-mp 5 . . . . . . . . . . 11 ((𝑢𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝})
5448simprd 499 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑣 “ {𝑝}) ⊆ 𝑦)
5553, 54sstrid 3886 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ 𝑦)
5650, 55ssind 4121 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥𝑦) ∧ (𝑢𝑈𝑣𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦))
57 imaeq1 5892 . . . . . . . . . . 11 (𝑤 = (𝑢𝑣) → (𝑤 “ {𝑝}) = ((𝑢𝑣) “ {𝑝}))
5857sseq1d 3906 . . . . . . . . . 10 (𝑤 = (𝑢𝑣) → ((𝑤 “ {𝑝}) ⊆ (𝑥𝑦) ↔ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)))
5958rspcev 3524 . . . . . . . . 9 (((𝑢𝑣) ∈ 𝑈 ∧ ((𝑢𝑣) “ {𝑝}) ⊆ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
6044, 56, 59syl2anc 587 . . . . . . . 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 767 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑈 ∈ (UnifOn‘𝑋))
64 simplrl 777 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑥 ∈ (unifTop‘𝑈))
65 simpr 488 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝 ∈ (𝑥𝑦))
66 elin 3857 . . . . . . . . . 10 (𝑝 ∈ (𝑥𝑦) ↔ (𝑝𝑥𝑝𝑦))
6765, 66sylib 221 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → (𝑝𝑥𝑝𝑦))
6867simpld 498 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑝𝑥)
6935simprd 499 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → ∀𝑝𝑥𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
7069r19.21bi 3120 . . . . . . . 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 778 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → 𝑦 ∈ (unifTop‘𝑈))
7367simprd 499 . . . . . . . 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 3269 . . . . . . 7 (∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦) ↔ (∃𝑢𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥 ∧ ∃𝑣𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
7671, 74, 75sylanbrc 586 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑢𝑈𝑣𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
7762, 76r19.29vva 3242 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥𝑦)) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
7877ralrimiva 3096 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))
79 elutop 22978 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8079adantr 484 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ((𝑥𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥𝑦)∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥𝑦))))
8139, 78, 80mpbir2and 713 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥𝑦) ∈ (unifTop‘𝑈))
8281ralrimivva 3103 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥𝑦) ∈ (unifTop‘𝑈))
83 fvex 6681 . . 3 (unifTop‘𝑈) ∈ V
84 istopg 21639 . . 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 586 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wcel 2113  wral 3053  wrex 3054  {crab 3057  Vcvv 3397  cin 3840  wss 3841  𝒫 cpw 4485  {csn 4513   cuni 4793  cima 5522  cfv 6333  Topctop 21637  UnifOncust 22944  unifTopcutop 22975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-fv 6341  df-top 21638  df-ust 22945  df-utop 22976
This theorem is referenced by:  utoptopon  22981  utop2nei  22995  utop3cls  22996  utopreg  22997
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