Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opnssborel Structured version   Visualization version   GIF version

Theorem opnssborel 39525
 Description: Open sets of a generalized real Euclidean space are Borel sets (notice that this theorem is even more general, because 𝑋 is not required to be a set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
opnssborel.a 𝐴 = (TopOpen‘(ℝ^‘𝑋))
opnssborel.b 𝐵 = (SalGen‘𝐴)
Assertion
Ref Expression
opnssborel 𝐴𝐵

Proof of Theorem opnssborel
StepHypRef Expression
1 opnssborel.a . . 3 𝐴 = (TopOpen‘(ℝ^‘𝑋))
2 fvex 6113 . . 3 (TopOpen‘(ℝ^‘𝑋)) ∈ V
31, 2eqeltri 2684 . 2 𝐴 ∈ V
4 opnssborel.b . . 3 𝐵 = (SalGen‘𝐴)
54sssalgen 39229 . 2 (𝐴 ∈ V → 𝐴𝐵)
63, 5ax-mp 5 1 𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ‘cfv 5804  TopOpenctopn 15905  ℝ^crrx 22979  SalGencsalgen 39208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-salg 39205  df-salgen 39209 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator