Theorem opnssborel 45809

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 6905	. 2 ⊢ 𝐴 ∈ V
3		opnssborel.b	. . 3 ⊢ 𝐵 = (SalGen‘𝐴)
4	3	sssalgen 45509	. 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵)
5	2, 4	ax-mp 5	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ⊆ wss 3948  ‘cfv 6543  TopOpenctopn 17374  ℝ^crrx 25230  SalGencsalgen 45486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-salg 45483  df-salgen 45487

This theorem is referenced by: (None)