Theorem dmsigagen 34543

Description: A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Assertion

Ref	Expression
dmsigagen	⊢ dom sigaGen = V

Proof of Theorem dmsigagen

Dummy variables 𝑗 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vuniex 7739	. . . . . 6 ⊢ ∪ 𝑗 ∈ V
2		pwsiga 34529	. . . . . 6 ⊢ (∪ 𝑗 ∈ V → 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗))
3	1, 2	ax-mp 5	. . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗)
4		pwuni 4910	. . . . 5 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗
5		sseq2 3962	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑗 → (𝑗 ⊆ 𝑠 ↔ 𝑗 ⊆ 𝒫 ∪ 𝑗))
6	5	rspcev 3580	. . . . 5 ⊢ ((𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) ∧ 𝑗 ⊆ 𝒫 ∪ 𝑗) → ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠)
7	3, 4, 6	mp2an 704	. . . 4 ⊢ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠
8		rabn0 4345	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠)
9	7, 8	mpbir 234	. . 3 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅
10		intex 5313	. . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V)
11	9, 10	mpbi 233	. 2 ⊢ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V
12		df-sigagen 34538	. 2 ⊢ sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠})
13	11, 12	dmmpti 6679	1 ⊢ dom sigaGen = V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088  {crab 3415  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  ∪ cuni 4871  ∩ cint 4911  dom cdm 5660  ‘cfv 6536  sigAlgebracsiga 34507  sigaGencsigagen 34537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-siga 34508  df-sigagen 34538

This theorem is used by: (None)