Theorem isrnsiga 33641

Description: The property of being a sigma-algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (Proof shortened by Thierry Arnoux, 23-Oct-2016.)

Assertion

Ref	Expression
isrnsiga	⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

Distinct variable group:   𝑥,𝑜,𝑆

Proof of Theorem isrnsiga

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-siga 33637	. . 3 ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
2		sigaex 33638	. . 3 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V
3		sseq1 4002	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑆 ⊆ 𝒫 𝑜))
4		eleq2 2816	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑜 ∈ 𝑠 ↔ 𝑜 ∈ 𝑆))
5		eleq2 2816	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑜 ∖ 𝑥) ∈ 𝑆))
6	5	raleqbi1dv 3327	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆))
7		pweq 4611	. . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
8		eleq2 2816	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
9	8	imbi2d 340	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
10	7, 9	raleqbidv 3336	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
11	4, 6, 10	3anbi123d 1432	. . . 4 ⊢ (𝑠 = 𝑆 → ((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))))
12	3, 11	anbi12d 630	. . 3 ⊢ (𝑠 = 𝑆 → ((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)))))
13	1, 2, 12	abfmpunirn 32386	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
14		rexv 3494	. . 3 ⊢ (∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) ↔ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
15	14	anbi2i 622	. 2 ⊢ ((𝑆 ∈ V ∧ ∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
16	13, 15	bitri 275	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064  Vcvv 3468   ∖ cdif 3940   ⊆ wss 3943  𝒫 cpw 4597  ∪ cuni 4902   class class class wbr 5141  ran crn 5670  ωcom 7852   ≼ cdom 8939  sigAlgebracsiga 33636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-siga 33637

This theorem is referenced by:  0elsiga  33642  sigaclcu  33645  issgon  33651