Theorem utoptopon 23962

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 23960	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
2		utopbas 23961	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
3		istopon 22635	. 2 ⊢ ((unifTop‘𝑈) ∈ (TopOn‘𝑋) ↔ ((unifTop‘𝑈) ∈ Top ∧ 𝑋 = ∪ (unifTop‘𝑈)))
4	1, 2, 3	sylanbrc 582	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∪ cuni 4908  ‘cfv 6543  Topctop 22616  TopOnctopon 22633  UnifOncust 23925  unifTopcutop 23956

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 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22617  df-topon 22634  df-ust 23926  df-utop 23957

This theorem is referenced by:  utop3cls  23977  tustps  23999