Theorem opnssborel 47407

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 6899	. 2 ⊢ 𝐴 ∈ V
3		opnssborel.b	. . 3 ⊢ 𝐵 = (SalGen‘𝐴)
4	3	sssalgen 47107	. 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵)
5	2, 4	ax-mp 5	1 ⊢ 𝐴 ⊆ 𝐵

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  ‘cfv 6540  TopOpenctopn 17496  ℝ^crrx 25593  SalGencsalgen 47084

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-salg 47081  df-salgen 47085

This theorem is used by: (None)