Theorem sigainb 34116

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 5324	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3		inss2 4245	. . 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 4624	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝒫 𝐴)
7	5, 6	syl 17	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 𝐴)
8	5, 7	elind 4209	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9		simpll 767	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ∈ ∪ ran sigAlgebra)
10		simplr 769	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴 ∈ 𝑆)
11		inss1 4244	. . . . . . 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 3992	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ 𝑆)
14		difelsiga 34113	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐴 ∖ 𝑥) ∈ 𝑆)
15	9, 10, 13, 14	syl3anc 1370	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝑆)
16		difss 4145	. . . . . . 7 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
17		elpwg 4607	. . . . . . 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 4209	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
21	20	ralrimiva 3143	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22		simplll 775	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
23		simplr 769	. . . . . . . 8 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24		elpwi 4611	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25		sstr 4003	. . . . . . . . . 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 4607	. . . . . . . . 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 34097	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
33	22, 30, 31, 32	syl3anc 1370	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
34		sstr 4003	. . . . . . . . 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 5104	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
38		vuniex 7757	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
39	38	elpw 4608	. . . . . . . 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 4209	. . . . 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 3143	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
45	8, 21, 44	3jca 1127	. 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 34092	. . 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 837	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2105  ∀wral 3058  Vcvv 3477   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  ∪ cuni 4911   class class class wbr 5147  ran crn 5689  ‘cfv 6562  ωcom 7886   ≼ cdom 8981  sigAlgebracsiga 34088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-ac2 10500

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-dju 9938  df-card 9976  df-acn 9979  df-ac 10153  df-siga 34089

This theorem is referenced by:  measinb2  34203