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Theorem sigainb 30199
Description: Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
sigainb |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( S i^i ~P A ) e. (sigAlgebra ` A ) )
Proof of Theorem sigainb
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 inex1g 4801 . . 3 |- ( S e. U. ran sigAlgebra -> ( S i^i ~P A ) e. _V )
21adantr 481 . 2 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( S i^i ~P A ) e. _V )
3 inss2 3834 . . 3 |- ( S i^i ~P A ) C_ ~P A
43a1i 11 . 2 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( S i^i ~P A ) C_ ~P A )
5 simpr 477 . . . 4 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A e. S )
6 pwidg 4173 . . . . 5 |- ( A e. S -> A e. ~P A )
75, 6syl 17 . . . 4 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A e. ~P A )
85, 7elind 3798 . . 3 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A e. ( S i^i ~P A ) )
9 simpll 790 . . . . . 6 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> S e. U. ran sigAlgebra )
10 simplr 792 . . . . . 6 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> A e. S )
11 inss1 3833 . . . . . . 7 |- ( S i^i ~P A ) C_ S
12 simpr 477 . . . . . . 7 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> x e. ( S i^i ~P A ) )
1311, 12sseldi 3601 . . . . . 6 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> x e. S )
14 difelsiga 30196 . . . . . 6 |- ( ( S e. U. ran sigAlgebra /\ A e. S /\ x e. S ) -> ( A \ x ) e. S )
159, 10, 13, 14syl3anc 1326 . . . . 5 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> ( A \ x ) e. S )
16 difss 3737 . . . . . . 7 |- ( A \ x ) C_ A
17 elpwg 4166 . . . . . . 7 |- ( ( A \ x ) e. S -> ( ( A \ x ) e. ~P A <-> ( A \ x ) C_ A ) )
1816, 17mpbiri 248 . . . . . 6 |- ( ( A \ x ) e. S -> ( A \ x ) e. ~P A )
1915, 18syl 17 . . . . 5 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> ( A \ x ) e. ~P A )
2015, 19elind 3798 . . . 4 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ( S i^i ~P A ) ) -> ( A \ x ) e. ( S i^i ~P A ) )
2120ralrimiva 2966 . . 3 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A. x e. ( S i^i ~P A ) ( A \ x ) e. ( S i^i ~P A ) )
22 simplll 798 . . . . . . 7 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> S e. U. ran sigAlgebra )
23 simplr 792 . . . . . . . 8 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> x e. ~P ( S i^i ~P A ) )
24 elpwi 4168 . . . . . . . . 9 |- ( x e. ~P ( S i^i ~P A ) -> x C_ ( S i^i ~P A ) )
25 sstr 3611 . . . . . . . . . 10 |- ( ( x C_ ( S i^i ~P A ) /\ ( S i^i ~P A ) C_ S ) -> x C_ S )
2611, 25mpan2 707 . . . . . . . . 9 |- ( x C_ ( S i^i ~P A ) -> x C_ S )
2723, 24, 263syl 18 . . . . . . . 8 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> x C_ S )
28 elpwg 4166 . . . . . . . . 9 |- ( x e. ~P ( S i^i ~P A ) -> ( x e. ~P S <-> x C_ S ) )
2928biimpar 502 . . . . . . . 8 |- ( ( x e. ~P ( S i^i ~P A ) /\ x C_ S ) -> x e. ~P S )
3023, 27, 29syl2anc 693 . . . . . . 7 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> x e. ~P S )
31 simpr 477 . . . . . . 7 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> x ~<_ om )
32 sigaclcu 30180 . . . . . . 7 |- ( ( S e. U. ran sigAlgebra /\ x e. ~P S /\ x ~<_ om ) -> U. x e. S )
3322, 30, 31, 32syl3anc 1326 . . . . . 6 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> U. x e. S )
34 sstr 3611 . . . . . . . . 9 |- ( ( x C_ ( S i^i ~P A ) /\ ( S i^i ~P A ) C_ ~P A ) -> x C_ ~P A )
353, 34mpan2 707 . . . . . . . 8 |- ( x C_ ( S i^i ~P A ) -> x C_ ~P A )
3623, 24, 353syl 18 . . . . . . 7 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> x C_ ~P A )
37 sspwuni 4611 . . . . . . . 8 |- ( x C_ ~P A <-> U. x C_ A )
38 vuniex 6954 . . . . . . . . 9 |- U. x e. _V
3938elpw 4164 . . . . . . . 8 |- ( U. x e. ~P A <-> U. x C_ A )
4037, 39bitr4i 267 . . . . . . 7 |- ( x C_ ~P A <-> U. x e. ~P A )
4136, 40sylib 208 . . . . . 6 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> U. x e. ~P A )
4233, 41elind 3798 . . . . 5 |- ( ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) /\ x ~<_ om ) -> U. x e. ( S i^i ~P A ) )
4342ex 450 . . . 4 |- ( ( ( S e. U. ran sigAlgebra /\ A e. S ) /\ x e. ~P ( S i^i ~P A ) ) -> ( x ~<_ om -> U. x e. ( S i^i ~P A ) ) )
4443ralrimiva 2966 . . 3 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A. x e. ~P ( S i^i ~P A ) ( x ~<_ om -> U. x e. ( S i^i ~P A ) ) )
458, 21, 443jca 1242 . 2 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( A e. ( S i^i ~P A ) /\ A. x e. ( S i^i ~P A ) ( A \ x ) e. ( S i^i ~P A ) /\ A. x e. ~P ( S i^i ~P A ) ( x ~<_ om -> U. x e. ( S i^i ~P A ) ) ) )
46 issiga 30174 . . 3 |- ( ( S i^i ~P A ) e. _V -> ( ( S i^i ~P A ) e. (sigAlgebra ` A ) <-> ( ( S i^i ~P A ) C_ ~P A /\ ( A e. ( S i^i ~P A ) /\ A. x e. ( S i^i ~P A ) ( A \ x ) e. ( S i^i ~P A ) /\ A. x e. ~P ( S i^i ~P A ) ( x ~<_ om -> U. x e. ( S i^i ~P A ) ) ) ) ) )
4746biimpar 502 . 2 |- ( ( ( S i^i ~P A ) e. _V /\ ( ( S i^i ~P A ) C_ ~P A /\ ( A e. ( S i^i ~P A ) /\ A. x e. ( S i^i ~P A ) ( A \ x ) e. ( S i^i ~P A ) /\ A. x e. ~P ( S i^i ~P A ) ( x ~<_ om -> U. x e. ( S i^i ~P A ) ) ) ) ) -> ( S i^i ~P A ) e. (sigAlgebra ` A ) )
482, 4, 45, 47syl12anc 1324 1 |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> ( S i^i ~P A ) e. (sigAlgebra ` A ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   ran crn 5115   ` cfv 5888   omcom 7065   ~<_ cdom 7953  sigAlgebracsiga 30170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-siga 30171
This theorem is referenced by:  measinb2  30286
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