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Theorem elunirn 5633
Description: Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
Assertion
Ref Expression
elunirn  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem elunirn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eluni 3707 . 2  |-  ( A  e.  U. ran  F  <->  E. y ( A  e.  y  /\  y  e. 
ran  F ) )
2 funfn 5121 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
3 fvelrnb 5435 . . . . . . . 8  |-  ( F  Fn  dom  F  -> 
( y  e.  ran  F  <->  E. x  e.  dom  F ( F `  x
)  =  y ) )
42, 3sylbi 120 . . . . . . 7  |-  ( Fun 
F  ->  ( y  e.  ran  F  <->  E. x  e.  dom  F ( F `
 x )  =  y ) )
54anbi2d 457 . . . . . 6  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <-> 
( A  e.  y  /\  E. x  e. 
dom  F ( F `
 x )  =  y ) ) )
6 r19.42v 2563 . . . . . 6  |-  ( E. x  e.  dom  F
( A  e.  y  /\  ( F `  x )  =  y )  <->  ( A  e.  y  /\  E. x  e.  dom  F ( F `
 x )  =  y ) )
75, 6syl6bbr 197 . . . . 5  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <->  E. x  e.  dom  F ( A  e.  y  /\  ( F `  x )  =  y ) ) )
8 eleq2 2179 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  ( A  e.  ( F `  x )  <->  A  e.  y ) )
98biimparc 295 . . . . . 6  |-  ( ( A  e.  y  /\  ( F `  x )  =  y )  ->  A  e.  ( F `  x ) )
109reximi 2504 . . . . 5  |-  ( E. x  e.  dom  F
( A  e.  y  /\  ( F `  x )  =  y )  ->  E. x  e.  dom  F  A  e.  ( F `  x
) )
117, 10syl6bi 162 . . . 4  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
1211exlimdv 1773 . . 3  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
13 fvelrn 5517 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
14 funfvex 5404 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
15 eleq2 2179 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  ( A  e.  y  <->  A  e.  ( F `  x ) ) )
16 eleq1 2178 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
y  e.  ran  F  <->  ( F `  x )  e.  ran  F ) )
1715, 16anbi12d 462 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
( A  e.  y  /\  y  e.  ran  F )  <->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
1817spcegv 2746 . . . . . 6  |-  ( ( F `  x )  e.  _V  ->  (
( A  e.  ( F `  x )  /\  ( F `  x )  e.  ran  F )  ->  E. y
( A  e.  y  /\  y  e.  ran  F ) ) )
1914, 18syl 14 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F )  ->  E. y
( A  e.  y  /\  y  e.  ran  F ) ) )
2013, 19mpan2d 422 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) ) )
2120rexlimdva 2524 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  dom  F  A  e.  ( F `  x
)  ->  E. y
( A  e.  y  /\  y  e.  ran  F ) ) )
2212, 21impbid 128 . 2  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
231, 22syl5bb 191 1  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   E.wrex 2392   _Vcvv 2658   U.cuni 3704   dom cdm 4507   ran crn 4508   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  fnunirn  5634
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