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Theorem iinexg 4360
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 4124 . . 3  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
21adantl 453 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
3 elisset 2966 . . . . . . . . 9  |-  ( B  e.  C  ->  E. y 
y  =  B )
43rgenw 2773 . . . . . . . 8  |-  A. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B )
5 r19.2z 3717 . . . . . . . 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 653 . . . . . . 7  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B ) )
7 r19.35 2855 . . . . . . 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 189 . . . . . 6  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  B  e.  C  ->  E. x  e.  A  E. y 
y  =  B ) )
98imp 419 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. x  e.  A  E. y 
y  =  B )
10 rexcom4 2975 . . . . 5  |-  ( E. x  e.  A  E. y  y  =  B  <->  E. y E. x  e.  A  y  =  B )
119, 10sylib 189 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. y E. x  e.  A  y  =  B )
12 abn0 3646 . . . 4  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  E. y E. x  e.  A  y  =  B )
1311, 12sylibr 204 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
14 intex 4356 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
1513, 14sylib 189 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
162, 15eqeltrd 2510 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 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706   _Vcvv 2956   (/)c0 3628   |^|cint 4050   |^|_ciin 4094
This theorem is referenced by:  fclsval  18040  taylfval  20275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-int 4051  df-iin 4096
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