Theorem sigagenval 34174

Description: Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
sigagenval	⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

Distinct variable group:   𝐴,𝑠

Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem sigagenval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sigagen 34173	. . 3 ⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠}))
3		unieq 4869	. . . . . 6 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
4	3	fveq2d 6832	. . . . 5 ⊢ (𝑥 = 𝐴 → (sigAlgebra‘∪ 𝑥) = (sigAlgebra‘∪ 𝐴))
5		sseq1 3956	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠))
6	4, 5	rabeqbidv 3414	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
7	6	inteqd 4902	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
8	7	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
9		elex 3458	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
10		uniexg 7679	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
11		pwsiga 34164	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
13		pwuni 4896	. . . . . 6 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
14	12, 13	jctir 520	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
15		sseq2 3957	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝐴 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴))
16	15	elrab 3643	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ↔ (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
17	14, 16	sylibr 234	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
18	17	ne0d 4291	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
19		intex 5284	. . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
20	18, 19	sylib 218	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
21	2, 8, 9, 20	fvmptd 6942	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  {crab 3396  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  ∪ cuni 4858  ∩ cint 4897   ↦ cmpt 5174  ‘cfv 6486  sigAlgebracsiga 34142  sigaGencsigagen 34172

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-siga 34143  df-sigagen 34173

This theorem is referenced by:  sigagensiga  34175  sssigagen  34179  sigagenss  34183