Theorem sigagenval 34297

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 34296	. . 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 4874	. . . . . 6 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
4	3	fveq2d 6838	. . . . 5 ⊢ (𝑥 = 𝐴 → (sigAlgebra‘∪ 𝑥) = (sigAlgebra‘∪ 𝐴))
5		sseq1 3959	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠))
6	4, 5	rabeqbidv 3417	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
7	6	inteqd 4907	. . 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 3461	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
10		uniexg 7685	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
11		pwsiga 34287	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
13		pwuni 4901	. . . . . 6 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
14	12, 13	jctir 520	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
15		sseq2 3960	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝐴 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴))
16	15	elrab 3646	. . . . 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 4294	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
19		intex 5289	. . 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 6948	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  {crab 3399  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ∩ cint 4902   ↦ cmpt 5179  ‘cfv 6492  sigAlgebracsiga 34265  sigaGencsigagen 34295

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-siga 34266  df-sigagen 34296

This theorem is referenced by:  sigagensiga  34298  sssigagen  34302  sigagenss  34306