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Theorem sigaclcuni 28566
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 4303 . . 3  |-  ( A. k  e.  A  B  e.  S  ->  U_ k  e.  A  B  =  U. { z  |  E. k  e.  A  z  =  B } )
213ad2ant2 1019 . 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 997 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  S  e.  U.
ran sigAlgebra )
4 r19.29 2942 . . . . . . . 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 459 . . . . . . . . . 10  |-  ( ( B  e.  S  /\  z  =  B )  ->  z  =  B )
6 simpl 455 . . . . . . . . . 10  |-  ( ( B  e.  S  /\  z  =  B )  ->  B  e.  S )
75, 6eqeltrd 2490 . . . . . . . . 9  |-  ( ( B  e.  S  /\  z  =  B )  ->  z  e.  S )
87rexlimivw 2893 . . . . . . . 8  |-  ( E. k  e.  A  ( B  e.  S  /\  z  =  B )  ->  z  e.  S )
94, 8syl 17 . . . . . . 7  |-  ( ( A. k  e.  A  B  e.  S  /\  E. k  e.  A  z  =  B )  -> 
z  e.  S )
109ex 432 . . . . . 6  |-  ( A. k  e.  A  B  e.  S  ->  ( E. k  e.  A  z  =  B  ->  z  e.  S ) )
1110abssdv 3513 . . . . 5  |-  ( A. k  e.  A  B  e.  S  ->  { z  |  E. k  e.  A  z  =  B }  C_  S )
12113ad2ant2 1019 . . . 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 4557 . . . . 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 17 . . . 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 27991 . . . 4  |-  ( A  ~<_  om  ->  { z  |  E. k  e.  A  z  =  B }  ~<_  om )
18173ad2ant3 1020 . . 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 28565 . . 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 1230 . 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 2490 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {cab 2387   F/_wnfc 2550   A.wral 2754   E.wrex 2755    C_ wss 3414   ~Pcpw 3955   U.cuni 4191   U_ciun 4271   class class class wbr 4395   ran crn 4824   omcom 6683    ~<_ cdom 7552  sigAlgebracsiga 28555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-ac2 8875
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-oi 7969  df-card 8352  df-acn 8355  df-ac 8529  df-siga 28556
This theorem is referenced by:  measvuni  28662  imambfm  28710  sibfof  28788
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