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Theorem sigainb 32735
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 ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

Proof of Theorem sigainb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 inex1g 5276 . . 3 (𝑆 ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 481 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 4189 . . 3 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
43a1i 11 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5 simpr 485 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴𝑆)
6 pwidg 4580 . . . . 5 (𝐴𝑆𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ 𝒫 𝐴)
85, 7elind 4154 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 765 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ran sigAlgebra)
10 simplr 767 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴𝑆)
11 inss1 4188 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12 simpr 485 . . . . . . 7 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sselid 3942 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥𝑆)
14 difelsiga 32732 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝑥𝑆) → (𝐴𝑥) ∈ 𝑆)
159, 10, 13, 14syl3anc 1371 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝑆)
16 difss 4091 . . . . . . 7 (𝐴𝑥) ⊆ 𝐴
17 elpwg 4563 . . . . . . 7 ((𝐴𝑥) ∈ 𝑆 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
1816, 17mpbiri 257 . . . . . 6 ((𝐴𝑥) ∈ 𝑆 → (𝐴𝑥) ∈ 𝒫 𝐴)
1915, 18syl 17 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝒫 𝐴)
2015, 19elind 4154 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3143 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 773 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ran sigAlgebra)
23 simplr 767 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 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 3952 . . . . . . . . . 10 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥𝑆)
2611, 25mpan2 689 . . . . . . . . 9 (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥𝑆)
2723, 24, 263syl 18 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
28 elpwg 4563 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆𝑥𝑆))
2928biimpar 478 . . . . . . . 8 ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥𝑆) → 𝑥 ∈ 𝒫 𝑆)
3023, 27, 29syl2anc 584 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31 simpr 485 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32 sigaclcu 32716 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆𝑥 ≼ ω) → 𝑥𝑆)
3322, 30, 31, 32syl3anc 1371 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
34 sstr 3952 . . . . . . . . 9 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
353, 34mpan2 689 . . . . . . . 8 (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
3623, 24, 353syl 18 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝒫 𝐴)
37 sspwuni 5060 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
38 vuniex 7676 . . . . . . . . 9 𝑥 ∈ V
3938elpw 4564 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
4037, 39bitr4i 277 . . . . . . 7 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
4136, 40sylib 217 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝐴)
4233, 41elind 4154 . . . . 5 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
4342ex 413 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
4443ralrimiva 3143 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
458, 21, 443jca 1128 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))
46 issiga 32711 . . 3 ((𝑆 ∩ 𝒫 𝐴) ∈ V → ((𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴) ↔ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))))
4746biimpar 478 . 2 (((𝑆 ∩ 𝒫 𝐴) ∈ V ∧ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
482, 4, 45, 47syl12anc 835 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865   class class class wbr 5105  ran crn 5634  cfv 6496  ωcom 7802  cdom 8881  sigAlgebracsiga 32707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-siga 32708
This theorem is referenced by:  measinb2  32822
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