Theorem sigaval 34135

Description: The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)

Assertion

Ref	Expression
sigaval	⊢ (𝑂 ∈ V → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})

Distinct variable group:   𝑥,𝑠,𝑂

Proof of Theorem sigaval

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3398	. . . 4 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))}
2		velpw 4556	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑂 ↔ 𝑠 ⊆ 𝒫 𝑂)
3	2	anbi1i 624	. . . . 5 ⊢ ((𝑠 ∈ 𝒫 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))) ↔ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))))
4	3	abbii 2800	. . . 4 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))}
5	1, 4	eqtri 2756	. . 3 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))}
6		pwexg 5320	. . . 4 ⊢ (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7		pwexg 5320	. . . 4 ⊢ (𝒫 𝑂 ∈ V → 𝒫 𝒫 𝑂 ∈ V)
8		rabexg 5279	. . . 4 ⊢ (𝒫 𝒫 𝑂 ∈ V → {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} ∈ V)
9	6, 7, 8	3syl 18	. . 3 ⊢ (𝑂 ∈ V → {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} ∈ V)
10	5, 9	eqeltrrid 2838	. 2 ⊢ (𝑂 ∈ V → {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V)
11		pweq 4565	. . . . . 6 ⊢ (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
12	11	sseq2d 3964	. . . . 5 ⊢ (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑂))
13		eleq1 2821	. . . . . 6 ⊢ (𝑜 = 𝑂 → (𝑜 ∈ 𝑠 ↔ 𝑂 ∈ 𝑠))
14		difeq1 4070	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
15	14	eleq1d 2818	. . . . . . 7 ⊢ (𝑜 = 𝑂 → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑠))
16	15	ralbidv 3157	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠))
17	13, 16	3anbi12d 1439	. . . . 5 ⊢ (𝑜 = 𝑂 → ((𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)) ↔ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))))
18	12, 17	anbi12d 632	. . . 4 ⊢ (𝑜 = 𝑂 → ((𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))) ↔ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))))
19	18	abbidv 2799	. . 3 ⊢ (𝑜 = 𝑂 → {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
20		df-siga 34133	. . 3 ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
21	19, 20	fvmptg 6936	. 2 ⊢ ((𝑂 ∈ V ∧ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V) → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
22	10, 21	mpdan 687	1 ⊢ (𝑂 ∈ V → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3049  {crab 3397  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  𝒫 cpw 4551  ∪ cuni 4860   class class class wbr 5095  ‘cfv 6489  ωcom 7805   ≼ cdom 8876  sigAlgebracsiga 34132

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-siga 34133

This theorem is referenced by: (None)