Theorem sigainb 34126

Description: Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)

Assertion

Ref	Expression
sigainb	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

Proof of Theorem sigainb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inex1g 5274	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3		inss2 4201	. . 3 ⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
4	3	a1i 11	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5		simpr 484	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
6		pwidg 4583	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝒫 𝐴)
7	5, 6	syl 17	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 𝐴)
8	5, 7	elind 4163	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9		simpll 766	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ∈ ∪ ran sigAlgebra)
10		simplr 768	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴 ∈ 𝑆)
11		inss1 4200	. . . . . . 7 ⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12		simpr 484	. . . . . . 7 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
13	11, 12	sselid 3944	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ 𝑆)
14		difelsiga 34123	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐴 ∖ 𝑥) ∈ 𝑆)
15	9, 10, 13, 14	syl3anc 1373	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝑆)
16		difss 4099	. . . . . . 7 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
17		elpwg 4566	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
18	16, 17	mpbiri 258	. . . . . 6 ⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
19	15, 18	syl 17	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
20	15, 19	elind 4163	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
21	20	ralrimiva 3125	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22		simplll 774	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
23		simplr 768	. . . . . . . 8 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24		elpwi 4570	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25		sstr 3955	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥 ⊆ 𝑆)
26	11, 25	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝑆)
27	23, 24, 26	3syl 18	. . . . . . . 8 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝑆)
28		elpwg 4566	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆))
29	28	biimpar 477	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥 ⊆ 𝑆) → 𝑥 ∈ 𝒫 𝑆)
30	23, 27, 29	syl2anc 584	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31		simpr 484	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32		sigaclcu 34107	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
33	22, 30, 31, 32	syl3anc 1373	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
34		sstr 3955	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
35	3, 34	mpan2 691	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
36	23, 24, 35	3syl 18	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝒫 𝐴)
37		sspwuni 5064	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
38		vuniex 7715	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
39	38	elpw 4567	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
40	37, 39	bitr4i 278	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ∈ 𝒫 𝐴)
41	36, 40	sylib 218	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝒫 𝐴)
42	33, 41	elind 4163	. . . . 5 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
43	42	ex 412	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
44	43	ralrimiva 3125	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
45	8, 21, 44	3jca 1128	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))
46		issiga 34102	. . 3 ⊢ ((𝑆 ∩ 𝒫 𝐴) ∈ V → ((𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴) ↔ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))))
47	46	biimpar 477	. 2 ⊢ (((𝑆 ∩ 𝒫 𝐴) ∈ V ∧ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
48	2, 4, 45, 47	syl12anc 836	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   class class class wbr 5107  ran crn 5639  ‘cfv 6511  ωcom 7842   ≼ cdom 8916  sigAlgebracsiga 34098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-ac2 10416

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-oi 9463  df-dju 9854  df-card 9892  df-acn 9895  df-ac 10069  df-siga 34099

This theorem is referenced by:  measinb2  34213