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Theorem sigainb 30524
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 4996 . . 3 (𝑆 ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 468 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 4030 . . 3 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
43a1i 11 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5 simpr 473 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴𝑆)
6 pwidg 4366 . . . . 5 (𝐴𝑆𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ 𝒫 𝐴)
85, 7elind 3997 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 774 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ran sigAlgebra)
10 simplr 776 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴𝑆)
11 inss1 4029 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12 simpr 473 . . . . . . 7 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sseldi 3796 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥𝑆)
14 difelsiga 30521 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝑥𝑆) → (𝐴𝑥) ∈ 𝑆)
159, 10, 13, 14syl3anc 1483 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝑆)
16 difss 3936 . . . . . . 7 (𝐴𝑥) ⊆ 𝐴
17 elpwg 4359 . . . . . . 7 ((𝐴𝑥) ∈ 𝑆 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
1816, 17mpbiri 249 . . . . . 6 ((𝐴𝑥) ∈ 𝑆 → (𝐴𝑥) ∈ 𝒫 𝐴)
1915, 18syl 17 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝒫 𝐴)
2015, 19elind 3997 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3154 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 782 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ran sigAlgebra)
23 simplr 776 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24 elpwi 4361 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25 sstr 3806 . . . . . . . . . 10 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥𝑆)
2611, 25mpan2 674 . . . . . . . . 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 4359 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆𝑥𝑆))
2928biimpar 465 . . . . . . . 8 ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥𝑆) → 𝑥 ∈ 𝒫 𝑆)
3023, 27, 29syl2anc 575 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31 simpr 473 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32 sigaclcu 30505 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆𝑥 ≼ ω) → 𝑥𝑆)
3322, 30, 31, 32syl3anc 1483 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
34 sstr 3806 . . . . . . . . 9 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
353, 34mpan2 674 . . . . . . . 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 4803 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
38 vuniex 7184 . . . . . . . . 9 𝑥 ∈ V
3938elpw 4357 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
4037, 39bitr4i 269 . . . . . . 7 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
4136, 40sylib 209 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝐴)
4233, 41elind 3997 . . . . 5 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
4342ex 399 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
4443ralrimiva 3154 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
458, 21, 443jca 1151 . 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 30499 . . 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 465 . 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 856 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wcel 2156  wral 3096  Vcvv 3391  cdif 3766  cin 3768  wss 3769  𝒫 cpw 4351   cuni 4630   class class class wbr 4844  ran crn 5312  cfv 6101  ωcom 7295  cdom 8190  sigAlgebracsiga 30495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-ac2 9570
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-oi 8654  df-card 9048  df-acn 9051  df-ac 9222  df-cda 9275  df-siga 30496
This theorem is referenced by:  measinb2  30611
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