Theorem sigaval 31979

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 3072	. . . 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 4535	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑂 ↔ 𝑠 ⊆ 𝒫 𝑂)
3	2	anbi1i 623	. . . . 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 2809	. . . 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 2766	. . 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 5296	. . . 4 ⊢ (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7		pwexg 5296	. . . 4 ⊢ (𝒫 𝑂 ∈ V → 𝒫 𝒫 𝑂 ∈ V)
8		rabexg 5250	. . . 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 2844	. 2 ⊢ (𝑂 ∈ V → {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V)
11		pweq 4546	. . . . . 6 ⊢ (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
12	11	sseq2d 3949	. . . . 5 ⊢ (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑂))
13		eleq1 2826	. . . . . 6 ⊢ (𝑜 = 𝑂 → (𝑜 ∈ 𝑠 ↔ 𝑂 ∈ 𝑠))
14		difeq1 4046	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
15	14	eleq1d 2823	. . . . . . 7 ⊢ (𝑜 = 𝑂 → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑠))
16	15	ralbidv 3120	. . . . . 6 ⊢ (𝑜 = 𝑂 → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠))
17	13, 16	3anbi12d 1435	. . . . 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 630	. . . 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 2808	. . 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 31977	. . 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 6855	. 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 683	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 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ‘cfv 6418  ωcom 7687   ≼ cdom 8689  sigAlgebracsiga 31976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-siga 31977

This theorem is referenced by: (None)