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Theorem elunirnALT 5795
Description: Membership in the union of the range of a function, proved directly. Unlike elunirn 5793, it doesn't appeal to ndmfv 5568 (via funiunfv 5790). (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 3846 . 2  |-  ( A  e.  U. ran  F  <->  E. y ( A  e.  y  /\  y  e. 
ran  F ) )
2 funfn 5299 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
3 fvelrnb 5586 . . . . . . . 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 2707 . . . . . 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 2357 . . . . . . 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 2663 . . . . 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 1626 . . 3  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
13 fvelrn 5677 . . . . . . 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 5555 . . . . . 6  |-  ( F `
 x )  e. 
_V
17 eleq2 2357 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( A  e.  y  <->  A  e.  ( F `  x ) ) )
18 eleq1 2356 . . . . . . 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 2888 . . . . 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 2680 . . 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 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   U.cuni 3843   dom cdm 4705   ran crn 4706   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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