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Theorem sigainb 29032
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 472 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  ( S  i^i  ~P A )  e.  _V )
3 inss2 3644 . . 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 468 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  A  e.  S )
6 pwidg 3955 . . . . 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 3609 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S )  ->  A  e.  ( S  i^i  ~P A ) )
9 simpll 768 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  S  e.  U. ran sigAlgebra )
10 simplr 770 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  A  e.  S )
11 inss1 3643 . . . . . . 7  |-  ( S  i^i  ~P A ) 
C_  S
12 simpr 468 . . . . . . 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 3416 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  x  e.  S )
14 difelsiga 29029 . . . . . 6  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  x  e.  S )  ->  ( A  \  x
)  e.  S )
159, 10, 13, 14syl3anc 1292 . . . . 5  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  ( A  \  x )  e.  S )
16 difss 3549 . . . . . . 7  |-  ( A 
\  x )  C_  A
17 elpwg 3950 . . . . . . 7  |-  ( ( A  \  x )  e.  S  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
1816, 17mpbiri 241 . . . . . 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 3609 . . . 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 2809 . . 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 776 . . . . . . 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 770 . . . . . . . 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 3951 . . . . . . . . 9  |-  ( x  e.  ~P ( S  i^i  ~P A )  ->  x  C_  ( S  i^i  ~P A ) )
25 sstr 3426 . . . . . . . . . 10  |-  ( ( x  C_  ( S  i^i  ~P A )  /\  ( S  i^i  ~P A
)  C_  S )  ->  x  C_  S )
2611, 25mpan2 685 . . . . . . . . 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 3950 . . . . . . . . 9  |-  ( x  e.  ~P ( S  i^i  ~P A )  ->  ( x  e. 
~P S  <->  x  C_  S
) )
2928biimpar 493 . . . . . . . 8  |-  ( ( x  e.  ~P ( S  i^i  ~P A )  /\  x  C_  S
)  ->  x  e.  ~P S )
3023, 27, 29syl2anc 673 . . . . . . 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 468 . . . . . . 7  |-  ( ( ( ( S  e. 
U. ran sigAlgebra  /\  A  e.  S )  /\  x  e.  ~P ( S  i^i  ~P A ) )  /\  x  ~<_  om )  ->  x  ~<_  om )
32 sigaclcu 29013 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  ~P S  /\  x  ~<_  om )  ->  U. x  e.  S
)
3322, 30, 31, 32syl3anc 1292 . . . . . 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 3426 . . . . . . . . 9  |-  ( ( x  C_  ( S  i^i  ~P A )  /\  ( S  i^i  ~P A
)  C_  ~P A
)  ->  x  C_  ~P A )
353, 34mpan2 685 . . . . . . . 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 4360 . . . . . . . 8  |-  ( x 
C_  ~P A  <->  U. x  C_  A )
38 vex 3034 . . . . . . . . . 10  |-  x  e. 
_V
3938uniex 6606 . . . . . . . . 9  |-  U. x  e.  _V
4039elpw 3948 . . . . . . . 8  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
4137, 40bitr4i 260 . . . . . . 7  |-  ( x 
C_  ~P A  <->  U. x  e.  ~P A )
4236, 41sylib 201 . . . . . 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 3609 . . . . 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 441 . . . 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 2809 . . 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 1210 . 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 29007 . . 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 493 . 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 1290 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 376    /\ w3a 1007    e. wcel 1904   A.wral 2756   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395   ran crn 4840   ` cfv 5589   omcom 6711    ~<_ cdom 7585  sigAlgebracsiga 29003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-ac2 8911
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-card 8391  df-acn 8394  df-ac 8565  df-cda 8616  df-siga 29004
This theorem is referenced by:  measinb2  29119
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