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Theorem iinexg 4602
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 4353 . . 3  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
21adantl 466 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
3 elisset 3119 . . . . . . . . 9  |-  ( B  e.  C  ->  E. y 
y  =  B )
43rgenw 2820 . . . . . . . 8  |-  A. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B )
5 r19.2z 3912 . . . . . . . 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 671 . . . . . . 7  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B ) )
7 r19.35 3003 . . . . . . 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 196 . . . . . 6  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  B  e.  C  ->  E. x  e.  A  E. y 
y  =  B ) )
98imp 429 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. x  e.  A  E. y 
y  =  B )
10 rexcom4 3128 . . . . 5  |-  ( E. x  e.  A  E. y  y  =  B  <->  E. y E. x  e.  A  y  =  B )
119, 10sylib 196 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. y E. x  e.  A  y  =  B )
12 abn0 3799 . . . 4  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  E. y E. x  e.  A  y  =  B )
1311, 12sylibr 212 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
14 intex 4598 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
1513, 14sylib 196 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
162, 15eqeltrd 2550 1  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2447    =/= wne 2657   A.wral 2809   E.wrex 2810   _Vcvv 3108   (/)c0 3780   |^|cint 4277   |^|_ciin 4321
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 1963  ax-ext 2440  ax-sep 4563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-v 3110  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3781  df-int 4278  df-iin 4323
This theorem is referenced by:  fclsval  20239  taylfval  22483
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