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Theorem sigainb 26601
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 4456 . . 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 3592 . . 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 3894 . . . . 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 3561 . . 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 3591 . . . . . . 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 3375 . . . . . 6  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  x  e.  S )
14 difelsiga 26598 . . . . . 6  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  x  e.  S )  ->  ( A  \  x
)  e.  S )
159, 10, 13, 14syl3anc 1218 . . . . 5  |-  ( ( ( S  e.  U. ran sigAlgebra  /\  A  e.  S
)  /\  x  e.  ( S  i^i  ~P A
) )  ->  ( A  \  x )  e.  S )
16 difss 3504 . . . . . . 7  |-  ( A 
\  x )  C_  A
17 elpwg 3889 . . . . . . 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 3561 . . . 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 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 3890 . . . . . . . . 9  |-  ( x  e.  ~P ( S  i^i  ~P A )  ->  x  C_  ( S  i^i  ~P A ) )
25 sstr 3385 . . . . . . . . . 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 3889 . . . . . . . . 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 26582 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  ~P S  /\  x  ~<_  om )  ->  U. x  e.  S
)
3322, 30, 31, 32syl3anc 1218 . . . . . 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 3385 . . . . . . . . 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 4277 . . . . . . . 8  |-  ( x 
C_  ~P A  <->  U. x  C_  A )
38 vex 2996 . . . . . . . . . 10  |-  x  e. 
_V
3938uniex 6397 . . . . . . . . 9  |-  U. x  e.  _V
4039elpw 3887 . . . . . . . 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 3561 . . . . 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 1168 . 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 26576 . . 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 1216 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 965    e. wcel 1756   A.wral 2736   _Vcvv 2993    \ cdif 3346    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   U.cuni 4112   class class class wbr 4313   ran crn 4862   ` cfv 5439   omcom 6497    ~<_ cdom 7329  sigAlgebracsiga 26572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-ac2 8653
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-oi 7745  df-card 8130  df-acn 8133  df-ac 8307  df-cda 8358  df-siga 26573
This theorem is referenced by:  measinb2  26659
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