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Theorem sigainb 31012
 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 5114 . . 3 (𝑆 ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 481 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 4126 . . 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 4468 . . . . 5 (𝐴𝑆𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ 𝒫 𝐴)
85, 7elind 4092 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 763 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ran sigAlgebra)
10 simplr 765 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴𝑆)
11 inss1 4125 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12 simpr 485 . . . . . . 7 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sseldi 3887 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥𝑆)
14 difelsiga 31009 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝑥𝑆) → (𝐴𝑥) ∈ 𝑆)
159, 10, 13, 14syl3anc 1364 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝑆)
16 difss 4029 . . . . . . 7 (𝐴𝑥) ⊆ 𝐴
17 elpwg 4461 . . . . . . 7 ((𝐴𝑥) ∈ 𝑆 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
1816, 17mpbiri 259 . . . . . 6 ((𝐴𝑥) ∈ 𝑆 → (𝐴𝑥) ∈ 𝒫 𝐴)
1915, 18syl 17 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝒫 𝐴)
2015, 19elind 4092 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3149 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 771 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ran sigAlgebra)
23 simplr 765 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24 elpwi 4463 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25 sstr 3897 . . . . . . . . . 10 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥𝑆)
2611, 25mpan2 687 . . . . . . . . 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 4461 . . . . . . . . 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 30993 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆𝑥 ≼ ω) → 𝑥𝑆)
3322, 30, 31, 32syl3anc 1364 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
34 sstr 3897 . . . . . . . . 9 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
353, 34mpan2 687 . . . . . . . 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 4921 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
38 vuniex 7324 . . . . . . . . 9 𝑥 ∈ V
3938elpw 4459 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
4037, 39bitr4i 279 . . . . . . 7 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
4136, 40sylib 219 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝐴)
4233, 41elind 4092 . . . . 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 3149 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
458, 21, 443jca 1121 . 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 30988 . . 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 833 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   ∈ wcel 2081  ∀wral 3105  Vcvv 3437   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745   class class class wbr 4962  ran crn 5444  ‘cfv 6225  ωcom 7436   ≼ cdom 8355  sigAlgebracsiga 30984 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-ac2 9731 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-oi 8820  df-dju 9176  df-card 9214  df-acn 9217  df-ac 9388  df-siga 30985 This theorem is referenced by:  measinb2  31099
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