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Theorem elunirn 6100
Description: Membership in the union of the range of a function. See elunirnALT 6101 for a shorter proof which uses ax-pow 4571. (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 4193 . 2  |-  ( A  e.  U. ran  F  <->  E. y ( A  e.  y  /\  y  e. 
ran  F ) )
2 funfn 5554 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
3 fvelrnb 5852 . . . . . . . 8  |-  ( F  Fn  dom  F  -> 
( y  e.  ran  F  <->  E. x  e.  dom  F ( F `  x
)  =  y ) )
42, 3sylbi 195 . . . . . . 7  |-  ( Fun 
F  ->  ( y  e.  ran  F  <->  E. x  e.  dom  F ( F `
 x )  =  y ) )
54anbi2d 702 . . . . . 6  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <-> 
( A  e.  y  /\  E. x  e. 
dom  F ( F `
 x )  =  y ) ) )
6 r19.42v 2961 . . . . . 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 263 . . . . 5  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <->  E. x  e.  dom  F ( A  e.  y  /\  ( F `  x )  =  y ) ) )
8 eleq2 2475 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  ( A  e.  ( F `  x )  <->  A  e.  y ) )
98biimparc 485 . . . . . 6  |-  ( ( A  e.  y  /\  ( F `  x )  =  y )  ->  A  e.  ( F `  x ) )
109reximi 2871 . . . . 5  |-  ( E. x  e.  dom  F
( A  e.  y  /\  ( F `  x )  =  y )  ->  E. x  e.  dom  F  A  e.  ( F `  x
) )
117, 10syl6bi 228 . . . 4  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
1211exlimdv 1745 . . 3  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
13 fvelrn 5958 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
1413a1d 25 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  ( F `  x )  e.  ran  F ) )
1514ancld 551 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
16 fvex 5815 . . . . . 6  |-  ( F `
 x )  e. 
_V
17 eleq2 2475 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( A  e.  y  <->  A  e.  ( F `  x ) ) )
18 eleq1 2474 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  e.  ran  F  <->  ( F `  x )  e.  ran  F ) )
1917, 18anbi12d 709 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( A  e.  y  /\  y  e.  ran  F )  <->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
2016, 19spcev 3150 . . . . 5  |-  ( ( A  e.  ( F `
 x )  /\  ( F `  x )  e.  ran  F )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) )
2115, 20syl6 31 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) ) )
2221rexlimdva 2895 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  dom  F  A  e.  ( F `  x
)  ->  E. y
( A  e.  y  /\  y  e.  ran  F ) ) )
2312, 22impbid 191 . 2  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
241, 23syl5bb 257 1  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   E.wrex 2754   U.cuni 4190   dom cdm 4942   ran crn 4943   Fun wfun 5519    Fn wfn 5520   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533
This theorem is referenced by:  fnunirn  6102  fin23lem30  8674  ustn0  20907  elrnust  20911  ustbas  20914  metuvalOLD  21236  metuval  21237  elunirn2  27812  metidval  28202  pstmval  28207  elunirnmbfm  28581  fourierdlem70  37309  fourierdlem71  37310  fourierdlem80  37319
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