Theorem sigainb 34839

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 5283	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
2	1	adantr 486	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3		inss2 4137	. . 3 ⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
4	3	a1i 11	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5		simpr 490	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
6		pwidg 4581	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝒫 𝐴)
7	5, 6	syl 18	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 𝐴)
8	5, 7	elind 4100	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9		simpll 779	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ∈ ∪ ran sigAlgebra)
10		simplr 781	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴 ∈ 𝑆)
11		inss1 4136	. . . . . . 7 ⊢ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12		simpr 490	. . . . . . 7 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
13	11, 12	sselid 3911	. . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ 𝑆)
14		difelsiga 34837	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐴 ∖ 𝑥) ∈ 𝑆)
15	9, 10, 13, 14	syl3anc 1399	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝑆)
16		difss 4075	. . . . . . 7 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
17		elpwg 4563	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
18	16, 17	mpbiri 261	. . . . . 6 ⊢ ((𝐴 ∖ 𝑥) ∈ 𝑆 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
19	15, 18	syl 18	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
20	15, 19	elind 4100	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
21	20	ralrimiva 3158	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 ∖ 𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22		simplll 787	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
23		simplr 781	. . . . . . . 8 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24		elpwi 4567	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25		sstr 3921	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥 ⊆ 𝑆)
26	11, 25	mpan2 704	. . . . . . . . 9 ⊢ (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝑆)
27	23, 24, 26	3syl 19	. . . . . . . 8 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝑆)
28		elpwg 4563	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆))
29	28	biimpar 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥 ⊆ 𝑆) → 𝑥 ∈ 𝒫 𝑆)
30	23, 27, 29	syl2anc 596	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31		simpr 490	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32		sigaclcu 34819	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
33	22, 30, 31, 32	syl3anc 1399	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑆)
34		sstr 3921	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
35	3, 34	mpan2 704	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
36	23, 24, 35	3syl 19	. . . . . . 7 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝒫 𝐴)
37		sspwuni 5063	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
38		vuniex 7780	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
39	38	elpw 4564	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
40	37, 39	bitr4i 281	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝐴 ↔ ∪ 𝑥 ∈ 𝒫 𝐴)
41	36, 40	sylib 221	. . . . . 6 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝒫 𝐴)
42	33, 41	elind 4100	. . . . 5 ⊢ ((((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
43	42	ex 418	. . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
44	43	ralrimiva 3158	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → ∪ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
45	8, 21, 44	3jca 1146	. 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 34814	. . 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 483	. 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 850	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  ran crn 5656  ‘cfv 6558  ωcom 7901   ≼ cdom 8999  sigAlgebracsiga 34810

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-oi 9531  df-dju 10009  df-card 10047  df-siga 34811

This theorem is used by:  measinb2  34926