Theorem sigagenval 34755

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 34754	. . 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 4878	. . . . . 6 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
4	3	fveq2d 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (sigAlgebra‘∪ 𝑥) = (sigAlgebra‘∪ 𝐴))
5		sseq1 3956	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠))
6	4, 5	rabeqbidv 3430	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
7	6	inteqd 4912	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
8	7	adantl 487	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
9		elex 3472	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
10		uniexg 7746	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
11		pwsiga 34744	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
12	10, 11	syl 18	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
13		pwuni 4906	. . . . . 6 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
14	12, 13	jctir 530	. . . . 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 3645	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ↔ (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
17	14, 16	sylibr 237	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
18	17	ne0d 4288	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
19		intex 5305	. . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
20	18, 19	sylib 221	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
21	2, 8, 9, 20	fvmptd 6993	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  ∩ cint 4907   ↦ cmpt 5186  ‘cfv 6531  sigAlgebracsiga 34722  sigaGencsigagen 34753

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-siga 34723  df-sigagen 34754

This theorem is used by:  sigagensiga  34756  sssigagen  34760  sigagenss  34764