Theorem opnssborel 43630

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 6670	. 2 ⊢ 𝐴 ∈ V
3		opnssborel.b	. . 3 ⊢ 𝐵 = (SalGen‘𝐴)
4	3	sssalgen 43331	. 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵)
5	2, 4	ax-mp 5	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ⊆ wss 3859  ‘cfv 6333  TopOpenctopn 16743  ℝ^crrx 24073  SalGencsalgen 43310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-iota 6292  df-fun 6335  df-fv 6341  df-salg 43307  df-salgen 43311

This theorem is referenced by: (None)