Theorem opnssborel 46556

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	⊢ 𝐴 ⊆ 𝐵

Proof of Theorem opnssborel

Step	Hyp	Ref	Expression
1		opnssborel.a	. . 3 ⊢ 𝐴 = (TopOpen‘(ℝ^‘𝑋))
2	1	fvexi 6934	. 2 ⊢ 𝐴 ∈ V
3		opnssborel.b	. . 3 ⊢ 𝐵 = (SalGen‘𝐴)
4	3	sssalgen 46256	. 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵)
5	2, 4	ax-mp 5	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ‘cfv 6573  TopOpenctopn 17481  ℝ^crrx 25436  SalGencsalgen 46233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-salg 46230  df-salgen 46234

This theorem is referenced by: (None)