Theorem opnssborel 47078

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 6841	. 2 ⊢ 𝐴 ∈ V
3		opnssborel.b	. . 3 ⊢ 𝐵 = (SalGen‘𝐴)
4	3	sssalgen 46778	. 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵)
5	2, 4	ax-mp 5	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883  ‘cfv 6485  TopOpenctopn 17375  ℝ^crrx 25368  SalGencsalgen 46755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-salg 46752  df-salgen 46756

This theorem is referenced by: (None)