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Theorem iinexg 4171
Description: The existence of an indexed union.  x is normally a free-variable parameter in  B, which should be read  B ( x ). (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
iinexg  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  e.  _V )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem iinexg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfiin2g 3936 . . 3  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
21adantl 452 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
3 elisset 2798 . . . . . . . . 9  |-  ( B  e.  C  ->  E. y 
y  =  B )
43rgenw 2610 . . . . . . . 8  |-  A. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B )
5 r19.2z 3543 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ( B  e.  C  ->  E. y  y  =  B ) )  ->  E. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B ) )
64, 5mpan2 652 . . . . . . 7  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B ) )
7 r19.35 2687 . . . . . . 7  |-  ( E. x  e.  A  ( B  e.  C  ->  E. y  y  =  B )  <->  ( A. x  e.  A  B  e.  C  ->  E. x  e.  A  E. y 
y  =  B ) )
86, 7sylib 188 . . . . . 6  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  B  e.  C  ->  E. x  e.  A  E. y 
y  =  B ) )
98imp 418 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. x  e.  A  E. y 
y  =  B )
10 rexcom4 2807 . . . . 5  |-  ( E. x  e.  A  E. y  y  =  B  <->  E. y E. x  e.  A  y  =  B )
119, 10sylib 188 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. y E. x  e.  A  y  =  B )
12 abn0 3473 . . . 4  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  E. y E. x  e.  A  y  =  B )
1311, 12sylibr 203 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
14 intex 4167 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
1513, 14sylib 188 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
162, 15eqeltrd 2357 1  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   (/)c0 3455   |^|cint 3862   |^|_ciin 3906
This theorem is referenced by:  fclsval  17703  taylfval  19738  inttop2  25515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863  df-iin 3908
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