Theorem sigainb 31397

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 5225	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
2	1	adantr 483	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3		inss2 4208	. . 3 ⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
4	3	a1i 11	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5		simpr 487	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
6		pwidg 4563	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝒫 𝐴)
7	5, 6	syl 17	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 𝐴)
8	5, 7	elind 4173	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9		simpll 765	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ∈ ∪ ran sigAlgebra)
10		simplr 767	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴 ∈ 𝑆)
11		inss1 4207	. . . . . . 7 ⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12		simpr 487	. . . . . . 7 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
13	11, 12	sseldi 3967	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ 𝑆)
14		difelsiga 31394	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐴 ∖ 𝑥) ∈ 𝑆)
15	9, 10, 13, 14	syl3anc 1367	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝑆)
16		difss 4110	. . . . . . 7 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
17		elpwg 4544	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
18	16, 17	mpbiri 260	. . . . . 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 4173	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
21	20	ralrimiva 3184	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22		simplll 773	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
23		simplr 767	. . . . . . . 8 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24		elpwi 4550	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25		sstr 3977	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥 ⊆ 𝑆)
26	11, 25	mpan2 689	. . . . . . . . 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 4544	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆))
29	28	biimpar 480	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥 ⊆ 𝑆) → 𝑥 ∈ 𝒫 𝑆)
30	23, 27, 29	syl2anc 586	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31		simpr 487	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32		sigaclcu 31378	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
33	22, 30, 31, 32	syl3anc 1367	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
34		sstr 3977	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
35	3, 34	mpan2 689	. . . . . . . 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 5024	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
38		vuniex 7467	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
39	38	elpw 4545	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
40	37, 39	bitr4i 280	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ∈ 𝒫 𝐴)
41	36, 40	sylib 220	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝒫 𝐴)
42	33, 41	elind 4173	. . . . 5 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
43	42	ex 415	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
44	43	ralrimiva 3184	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
45	8, 21, 44	3jca 1124	. 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 31373	. . 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 480	. 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 834	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840   class class class wbr 5068  ran crn 5558  ‘cfv 6357  ωcom 7582   ≼ cdom 8509  sigAlgebracsiga 31369

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-ac2 9887

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-oi 8976  df-dju 9332  df-card 9370  df-acn 9373  df-ac 9544  df-siga 31370

This theorem is referenced by:  measinb2  31484