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Theorem fvelrnb 5253
Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
Assertion
Ref Expression
fvelrnb  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fvelrnb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-rex 2355 . . . 4  |-  ( E. x  e.  A  ( F `  x )  =  B  <->  E. x
( x  e.  A  /\  ( F `  x
)  =  B ) )
2 19.41v 1824 . . . . 5  |-  ( E. x ( ( x  e.  A  /\  ( F `  x )  =  B )  /\  F  Fn  A )  <->  ( E. x ( x  e.  A  /\  ( F `
 x )  =  B )  /\  F  Fn  A ) )
3 simpl 107 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  ( F `  x )  =  B )  ->  x  e.  A )
43anim1i 333 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( x  e.  A  /\  F  Fn  A ) )
54ancomd 263 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( F  Fn  A  /\  x  e.  A ) )
6 funfvex 5223 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
76funfni 5030 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
85, 7syl 14 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( F `  x )  e.  _V )
9 simpr 108 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ( F `  x )  =  B )  -> 
( F `  x
)  =  B )
109eleq1d 2148 . . . . . . . 8  |-  ( ( x  e.  A  /\  ( F `  x )  =  B )  -> 
( ( F `  x )  e.  _V  <->  B  e.  _V ) )
1110adantr 270 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( ( F `  x )  e.  _V  <->  B  e.  _V ) )
128, 11mpbid 145 . . . . . 6  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  B  e.  _V )
1312exlimiv 1530 . . . . 5  |-  ( E. x ( ( x  e.  A  /\  ( F `  x )  =  B )  /\  F  Fn  A )  ->  B  e.  _V )
142, 13sylbir 133 . . . 4  |-  ( ( E. x ( x  e.  A  /\  ( F `  x )  =  B )  /\  F  Fn  A )  ->  B  e.  _V )
151, 14sylanb 278 . . 3  |-  ( ( E. x  e.  A  ( F `  x )  =  B  /\  F  Fn  A )  ->  B  e.  _V )
1615expcom 114 . 2  |-  ( F  Fn  A  ->  ( E. x  e.  A  ( F `  x )  =  B  ->  B  e.  _V ) )
17 fnrnfv 5252 . . . 4  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
1817eleq2d 2149 . . 3  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  B  e.  { y  |  E. x  e.  A  y  =  ( F `  x ) } ) )
19 eqeq1 2088 . . . . . 6  |-  ( y  =  B  ->  (
y  =  ( F `
 x )  <->  B  =  ( F `  x ) ) )
20 eqcom 2084 . . . . . 6  |-  ( B  =  ( F `  x )  <->  ( F `  x )  =  B )
2119, 20syl6bb 194 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F `
 x )  <->  ( F `  x )  =  B ) )
2221rexbidv 2370 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  ( F `  x )  =  B ) )
2322elab3g 2745 . . 3  |-  ( ( E. x  e.  A  ( F `  x )  =  B  ->  B  e.  _V )  ->  ( B  e.  { y  |  E. x  e.  A  y  =  ( F `  x ) }  <->  E. x  e.  A  ( F `  x )  =  B ) )
2418, 23sylan9bbr 451 . 2  |-  ( ( ( E. x  e.  A  ( F `  x )  =  B  ->  B  e.  _V )  /\  F  Fn  A
)  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
2516, 24mpancom 413 1  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   E.wrex 2350   _Vcvv 2602   ran crn 4372    Fn wfn 4927   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940
This theorem is referenced by:  chfnrn  5310  rexrn  5336  ralrn  5337  elrnrexdmb  5339  ffnfv  5355  fconstfvm  5411  elunirn  5437  isoini  5488  reldm  5843  ordiso2  6505  uzn0  8715  frec2uzrand  9487  frecuzrdgtcl  9494  frecuzrdgfunlem  9501  uzin2  10011
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