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Theorem sigainb 27804
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 4590 . . 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 3719 . . 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 4023 . . . . 5  |-  ( A  e.  S  ->  A  e.  ~P A )
75, 6syl 16 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  A  e.  ~P A
)
85, 7elind 3688 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  A  e.  ( S  i^i  ~P A ) )
9 simpll 753 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  S  e.  U. ran sigAlgebra )
10 simplr 754 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  A  e.  S )
11 inss1 3718 . . . . . . 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 3502 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  x  e.  S )
14 difelsiga 27801 . . . . . 6  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  x  e.  S )  ->  ( A  \  x
)  e.  S )
159, 10, 13, 14syl3anc 1228 . . . . 5  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  ( A  \  x )  e.  S )
16 difss 3631 . . . . . . 7  |-  ( A 
\  x )  C_  A
17 elpwg 4018 . . . . . . 7  |-  ( ( A  \  x )  e.  S  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
1816, 17mpbiri 233 . . . . . 6  |-  ( ( A  \  x )  e.  S  ->  ( A  \  x )  e. 
~P A )
1915, 18syl 16 . . . . 5  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  ( A  \  x )  e. 
~P A )
2015, 19elind 3688 . . . 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 2878 . . 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 757 . . . . . . 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 754 . . . . . . . 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 4019 . . . . . . . . 9  |-  ( x  e.  ~P ( S  i^i  ~P A )  ->  x  C_  ( S  i^i  ~P A ) )
25 sstr 3512 . . . . . . . . . 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 20 . . . . . . . 8  |-  ( ( ( ( S  e. 
U. ran sigAlgebra  /\  A  e.  S )  /\  x  e.  ~P ( S  i^i  ~P A ) )  /\  x  ~<_  om )  ->  x  C_  S )
28 elpwg 4018 . . . . . . . . 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 27785 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  ~P S  /\  x  ~<_  om )  ->  U. x  e.  S
)
3322, 30, 31, 32syl3anc 1228 . . . . . 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 3512 . . . . . . . . 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 20 . . . . . . 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 4411 . . . . . . . 8  |-  ( x 
C_  ~P A  <->  U. x  C_  A )
38 vex 3116 . . . . . . . . . 10  |-  x  e. 
_V
3938uniex 6580 . . . . . . . . 9  |-  U. x  e.  _V
4039elpw 4016 . . . . . . . 8  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
4137, 40bitr4i 252 . . . . . . 7  |-  ( x 
C_  ~P A  <->  U. x  e.  ~P A )
4236, 41sylib 196 . . . . . 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 3688 . . . . 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 2878 . . 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 1176 . 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 27779 . . 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 1226 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 973    e. wcel 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   ran crn 5000   ` cfv 5588   omcom 6684    ~<_ cdom 7514  sigAlgebracsiga 27775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-ac2 8843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-card 8320  df-acn 8323  df-ac 8497  df-cda 8548  df-siga 27776
This theorem is referenced by:  measinb2  27862
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