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Theorem iinexg 4556
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 4306 . . 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 3072 . . . . . . . . 9  |-  ( B  e.  C  ->  E. y 
y  =  B )
43rgenw 2767 . . . . . . . 8  |-  A. x  e.  A  ( B  e.  C  ->  E. y 
y  =  B )
5 r19.2z 3864 . . . . . . . 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 2956 . . . . . . 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 198 . . . . . 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 3081 . . . . 5  |-  ( E. x  e.  A  E. y  y  =  B  <->  E. y E. x  e.  A  y  =  B )
119, 10sylib 198 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  E. y E. x  e.  A  y  =  B )
12 abn0 3760 . . . 4  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  E. y E. x  e.  A  y  =  B )
1311, 12sylibr 214 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
14 intex 4552 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  =/=  (/)  <->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
1513, 14sylib 198 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  _V )
162, 15eqeltrd 2492 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 1407   E.wex 1635    e. wcel 1844   {cab 2389    =/= wne 2600   A.wral 2756   E.wrex 2757   _Vcvv 3061   (/)c0 3740   |^|cint 4229   |^|_ciin 4274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-v 3063  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3741  df-int 4230  df-iin 4276
This theorem is referenced by:  fclsval  20803  taylfval  23048
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