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

Theorem opnssborel 46591
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))
21fvexi 6921 . 2 𝐴 ∈ V
3 opnssborel.b . . 3 𝐵 = (SalGen‘𝐴)
43sssalgen 46291 . 2 (𝐴 ∈ V → 𝐴𝐵)
52, 4ax-mp 5 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  cfv 6563  TopOpenctopn 17468  ℝ^crrx 25431  SalGencsalgen 46268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-salg 46265  df-salgen 46269
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator