Theorem utoptopon 23460

Description: Topology induced by a uniform structure 𝑈 with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)

Assertion

Ref	Expression
utoptopon	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))

Proof of Theorem utoptopon

Step	Hyp	Ref	Expression
1		utoptop 23458	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
2		utopbas 23459	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
3		istopon 22133	. 2 ⊢ ((unifTop‘𝑈) ∈ (TopOn‘𝑋) ↔ ((unifTop‘𝑈) ∈ Top ∧ 𝑋 = ∪ (unifTop‘𝑈)))
4	1, 2, 3	sylanbrc 583	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∪ cuni 4850  ‘cfv 6465  Topctop 22114  TopOnctopon 22131  UnifOncust 23423  unifTopcutop 23454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-fv 6473  df-top 22115  df-topon 22132  df-ust 23424  df-utop 23455

This theorem is referenced by:  utop3cls  23475  tustps  23497