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Theorem sigaclcuni 26699
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 1010 . 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 988 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  S  e.  U.
ran sigAlgebra )
4 r19.29 2956 . . . . . . . 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 2539 . . . . . . . . 9  |-  ( ( B  e.  S  /\  z  =  B )  ->  z  e.  S )
87rexlimivw 2936 . . . . . . . 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 3527 . . . . 5  |-  ( A. k  e.  A  B  e.  S  ->  { z  |  E. k  e.  A  z  =  B }  C_  S )
12113ad2ant2 1010 . . . 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 4556 . . . . 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 26166 . . . 4  |-  ( A  ~<_  om  ->  { z  |  E. k  e.  A  z  =  B }  ~<_  om )
18173ad2ant3 1011 . . 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 26698 . . 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 1219 . 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 2539 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 965    = wceq 1370    e. wcel 1758   {cab 2436   F/_wnfc 2599   A.wral 2795   E.wrex 2796    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   U_ciun 4272   class class class wbr 4393   ran crn 4942   omcom 6579    ~<_ cdom 7411  sigAlgebracsiga 26688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-ac2 8736
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-oi 7828  df-card 8213  df-acn 8216  df-ac 8390  df-siga 26689
This theorem is referenced by:  measvuni  26766  imambfm  26814  sibfof  26863
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