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Theorem sigainb 28597
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 4539 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  ( S  i^i  ~P A )  e.  _V )
21adantr 465 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  ( S  i^i  ~P A )  e.  _V )
3 inss2 3662 . . 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 461 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  A  e.  S )
6 pwidg 3970 . . . . 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 3629 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  A  e.  ( S  i^i  ~P A ) )
9 simpll 754 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  S  e.  U. ran sigAlgebra )
10 simplr 756 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  A  e.  S )
11 inss1 3661 . . . . . . 7  |-  ( S  i^i  ~P A ) 
C_  S
12 simpr 461 . . . . . . 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 3442 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  x  e.  S )
14 difelsiga 28594 . . . . . 6  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  x  e.  S )  ->  ( A  \  x
)  e.  S )
159, 10, 13, 14syl3anc 1232 . . . . 5  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  ( A  \  x )  e.  S )
16 difss 3572 . . . . . . 7  |-  ( A 
\  x )  C_  A
17 elpwg 3965 . . . . . . 7  |-  ( ( A  \  x )  e.  S  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
1816, 17mpbiri 235 . . . . . 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 3629 . . . 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 2820 . . 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 762 . . . . . . 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 756 . . . . . . . 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 3966 . . . . . . . . 9  |-  ( x  e.  ~P ( S  i^i  ~P A )  ->  x  C_  ( S  i^i  ~P A ) )
25 sstr 3452 . . . . . . . . . 10  |-  ( ( x  C_  ( S  i^i  ~P A )  /\  ( S  i^i  ~P A
)  C_  S )  ->  x  C_  S )
2611, 25mpan2 671 . . . . . . . . 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 3965 . . . . . . . . 9  |-  ( x  e.  ~P ( S  i^i  ~P A )  ->  ( x  e. 
~P S  <->  x  C_  S
) )
2928biimpar 485 . . . . . . . 8  |-  ( ( x  e.  ~P ( S  i^i  ~P A )  /\  x  C_  S
)  ->  x  e.  ~P S )
3023, 27, 29syl2anc 661 . . . . . . 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 461 . . . . . . 7  |-  ( ( ( ( S  e. 
U. ran sigAlgebra  /\  A  e.  S )  /\  x  e.  ~P ( S  i^i  ~P A ) )  /\  x  ~<_  om )  ->  x  ~<_  om )
32 sigaclcu 28578 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  ~P S  /\  x  ~<_  om )  ->  U. x  e.  S
)
3322, 30, 31, 32syl3anc 1232 . . . . . 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 3452 . . . . . . . . 9  |-  ( ( x  C_  ( S  i^i  ~P A )  /\  ( S  i^i  ~P A
)  C_  ~P A
)  ->  x  C_  ~P A )
353, 34mpan2 671 . . . . . . . 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 4362 . . . . . . . 8  |-  ( x 
C_  ~P A  <->  U. x  C_  A )
38 vex 3064 . . . . . . . . . 10  |-  x  e. 
_V
3938uniex 6580 . . . . . . . . 9  |-  U. x  e.  _V
4039elpw 3963 . . . . . . . 8  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
4137, 40bitr4i 254 . . . . . . 7  |-  ( x 
C_  ~P A  <->  U. x  e.  ~P A )
4236, 41sylib 198 . . . . . 6  |-  ( ( ( ( S  e. 
U. ran sigAlgebra  /\  A  e.  S )  /\  x  e.  ~P ( S  i^i  ~P A ) )  /\  x  ~<_  om )  ->  U. x  e.  ~P A )
4333, 42elind 3629 . . . . 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 ) )
4443ex 434 . . . 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 ) ) )
4544ralrimiva 2820 . . 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
) ) )
468, 21, 453jca 1179 . 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 ) ) ) )
47 issiga 28572 . . 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 ) ) ) ) ) )
4847biimpar 485 . 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 ) )
492, 4, 46, 48syl12anc 1230 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 369    /\ w3a 976    e. wcel 1844   A.wral 2756   _Vcvv 3061    \ cdif 3413    i^i cin 3415    C_ wss 3416   ~Pcpw 3957   U.cuni 4193   class class class wbr 4397   ran crn 4826   ` cfv 5571   omcom 6685    ~<_ cdom 7554  sigAlgebracsiga 28568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-ac2 8877
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-oi 7971  df-card 8354  df-acn 8357  df-ac 8531  df-cda 8582  df-siga 28569
This theorem is referenced by:  measinb2  28684
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