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Theorem elunirnALT 5779
Description: Membership in the union of the range of a function, proved directly. Unlike elunirn 5777, it doesn't appeal to ndmfv 5552 (via funiunfv 5774). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elunirnALT  |-  ( 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 elunirnALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eluni 3830 . 2  |-  ( A  e.  U. ran  F  <->  E. y ( A  e.  y  /\  y  e. 
ran  F ) )
2 funfn 5283 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
3 fvelrnb 5570 . . . . . . . 8  |-  ( F  Fn  dom  F  -> 
( y  e.  ran  F  <->  E. x  e.  dom  F ( F `  x
)  =  y ) )
42, 3sylbi 187 . . . . . . 7  |-  ( Fun 
F  ->  ( y  e.  ran  F  <->  E. x  e.  dom  F ( F `
 x )  =  y ) )
54anbi2d 684 . . . . . 6  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <-> 
( A  e.  y  /\  E. x  e. 
dom  F ( F `
 x )  =  y ) ) )
6 r19.42v 2694 . . . . . 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 254 . . . . 5  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <->  E. x  e.  dom  F ( A  e.  y  /\  ( F `  x )  =  y ) ) )
8 eleq2 2344 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  ( A  e.  ( F `  x )  <->  A  e.  y ) )
98biimparc 473 . . . . . 6  |-  ( ( A  e.  y  /\  ( F `  x )  =  y )  ->  A  e.  ( F `  x ) )
109reximi 2650 . . . . 5  |-  ( E. x  e.  dom  F
( A  e.  y  /\  ( F `  x )  =  y )  ->  E. x  e.  dom  F  A  e.  ( F `  x
) )
117, 10syl6bi 219 . . . 4  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
1211exlimdv 1664 . . 3  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
13 fvelrn 5661 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
1413a1d 22 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  ( F `  x )  e.  ran  F ) )
1514ancld 536 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
16 fvex 5539 . . . . . 6  |-  ( F `
 x )  e. 
_V
17 eleq2 2344 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( A  e.  y  <->  A  e.  ( F `  x ) ) )
18 eleq1 2343 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  e.  ran  F  <->  ( F `  x )  e.  ran  F ) )
1917, 18anbi12d 691 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( A  e.  y  /\  y  e.  ran  F )  <->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
2016, 19spcev 2875 . . . . 5  |-  ( ( A  e.  ( F `
 x )  /\  ( F `  x )  e.  ran  F )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) )
2115, 20syl6 29 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) ) )
2221rexlimdva 2667 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  dom  F  A  e.  ( F `  x
)  ->  E. y
( A  e.  y  /\  y  e.  ran  F ) ) )
2312, 22impbid 183 . 2  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
241, 23syl5bb 248 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    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   U.cuni 3827   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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