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Theorem fvelrnb 5435
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 2397 . . . 4  |-  ( E. x  e.  A  ( F `  x )  =  B  <->  E. x
( x  e.  A  /\  ( F `  x
)  =  B ) )
2 19.41v 1856 . . . . 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 108 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  ( F `  x )  =  B )  ->  x  e.  A )
43anim1i 336 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( x  e.  A  /\  F  Fn  A ) )
54ancomd 265 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( F  Fn  A  /\  x  e.  A ) )
6 funfvex 5404 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
76funfni 5191 . . . . . . . 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 109 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ( F `  x )  =  B )  -> 
( F `  x
)  =  B )
109eleq1d 2184 . . . . . . . 8  |-  ( ( x  e.  A  /\  ( F `  x )  =  B )  -> 
( ( F `  x )  e.  _V  <->  B  e.  _V ) )
1110adantr 272 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  ( ( F `  x )  e.  _V  <->  B  e.  _V ) )
128, 11mpbid 146 . . . . . 6  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =  B )  /\  F  Fn  A
)  ->  B  e.  _V )
1312exlimiv 1560 . . . . 5  |-  ( E. x ( ( x  e.  A  /\  ( F `  x )  =  B )  /\  F  Fn  A )  ->  B  e.  _V )
142, 13sylbir 134 . . . 4  |-  ( ( E. x ( x  e.  A  /\  ( F `  x )  =  B )  /\  F  Fn  A )  ->  B  e.  _V )
151, 14sylanb 280 . . 3  |-  ( ( E. x  e.  A  ( F `  x )  =  B  /\  F  Fn  A )  ->  B  e.  _V )
1615expcom 115 . 2  |-  ( F  Fn  A  ->  ( E. x  e.  A  ( F `  x )  =  B  ->  B  e.  _V ) )
17 fnrnfv 5434 . . . 4  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
1817eleq2d 2185 . . 3  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  B  e.  { y  |  E. x  e.  A  y  =  ( F `  x ) } ) )
19 eqeq1 2122 . . . . . 6  |-  ( y  =  B  ->  (
y  =  ( F `
 x )  <->  B  =  ( F `  x ) ) )
20 eqcom 2117 . . . . . 6  |-  ( B  =  ( F `  x )  <->  ( F `  x )  =  B )
2119, 20syl6bb 195 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F `
 x )  <->  ( F `  x )  =  B ) )
2221rexbidv 2413 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  ( F `  x )  =  B ) )
2322elab3g 2806 . . 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 456 . 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 416 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 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   E.wrex 2392   _Vcvv 2658   ran crn 4508    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:  chfnrn  5497  rexrn  5523  ralrn  5524  elrnrexdmb  5526  ffnfv  5544  fconstfvm  5604  elunirn  5633  isoini  5685  reldm  6050  ordiso2  6886  eldju  6919  ctssdc  6964  uzn0  9290  frec2uzrand  10118  frecuzrdgtcl  10125  frecuzrdgfunlem  10132  uzin2  10699
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