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Theorem sigaclcuni 27744
Description: A sigma-algebra is closed under countable union: indexed union version (Contributed by Thierry Arnoux, 8-Jun-2017.)
Hypothesis
Ref Expression
sigaclcuni.1  |-  F/_ k A
Assertion
Ref Expression
sigaclcuni  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  U_ k  e.  A  B  e.  S
)
Distinct variable group:    S, k
Allowed substitution hints:    A( k)    B( k)

Proof of Theorem sigaclcuni
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfiun2g 4350 . . 3  |-  ( A. k  e.  A  B  e.  S  ->  U_ k  e.  A  B  =  U. { z  |  E. k  e.  A  z  =  B } )
213ad2ant2 1013 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  U_ k  e.  A  B  =  U. { z  |  E. k  e.  A  z  =  B } )
3 simp1 991 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  S  e.  U.
ran sigAlgebra )
4 r19.29 2990 . . . . . . . 8  |-  ( ( A. k  e.  A  B  e.  S  /\  E. k  e.  A  z  =  B )  ->  E. k  e.  A  ( B  e.  S  /\  z  =  B
) )
5 simpr 461 . . . . . . . . . 10  |-  ( ( B  e.  S  /\  z  =  B )  ->  z  =  B )
6 simpl 457 . . . . . . . . . 10  |-  ( ( B  e.  S  /\  z  =  B )  ->  B  e.  S )
75, 6eqeltrd 2548 . . . . . . . . 9  |-  ( ( B  e.  S  /\  z  =  B )  ->  z  e.  S )
87rexlimivw 2945 . . . . . . . 8  |-  ( E. k  e.  A  ( B  e.  S  /\  z  =  B )  ->  z  e.  S )
94, 8syl 16 . . . . . . 7  |-  ( ( A. k  e.  A  B  e.  S  /\  E. k  e.  A  z  =  B )  -> 
z  e.  S )
109ex 434 . . . . . 6  |-  ( A. k  e.  A  B  e.  S  ->  ( E. k  e.  A  z  =  B  ->  z  e.  S ) )
1110abssdv 3567 . . . . 5  |-  ( A. k  e.  A  B  e.  S  ->  { z  |  E. k  e.  A  z  =  B }  C_  S )
12113ad2ant2 1013 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  { z  |  E. k  e.  A  z  =  B }  C_  S )
13 elpw2g 4603 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  ->  ( { z  |  E. k  e.  A  z  =  B }  e.  ~P S 
<->  { z  |  E. k  e.  A  z  =  B }  C_  S
) )
143, 13syl 16 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  ( {
z  |  E. k  e.  A  z  =  B }  e.  ~P S 
<->  { z  |  E. k  e.  A  z  =  B }  C_  S
) )
1512, 14mpbird 232 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  { z  |  E. k  e.  A  z  =  B }  e.  ~P S )
16 sigaclcuni.1 . . . . 5  |-  F/_ k A
1716abrexctf 27203 . . . 4  |-  ( A  ~<_  om  ->  { z  |  E. k  e.  A  z  =  B }  ~<_  om )
18173ad2ant3 1014 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  { z  |  E. k  e.  A  z  =  B }  ~<_  om )
19 sigaclcu 27743 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  { z  |  E. k  e.  A  z  =  B }  e.  ~P S  /\  { z  |  E. k  e.  A  z  =  B }  ~<_  om )  ->  U. {
z  |  E. k  e.  A  z  =  B }  e.  S
)
203, 15, 18, 19syl3anc 1223 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  U. { z  |  E. k  e.  A  z  =  B }  e.  S )
212, 20eqeltrd 2548 1  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  U_ k  e.  A  B  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {cab 2445   F/_wnfc 2608   A.wral 2807   E.wrex 2808    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   U_ciun 4318   class class class wbr 4440   ran crn 4993   omcom 6671    ~<_ cdom 7504  sigAlgebracsiga 27733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-ac2 8832
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-card 8309  df-acn 8312  df-ac 8486  df-siga 27734
This theorem is referenced by:  measvuni  27811  imambfm  27859  sibfof  27908
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