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Theorem fvrn0 5550
Description: A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
Assertion
Ref Expression
fvrn0  |-  ( F `
 X )  e.  ( ran  F  u.  {
(/) } )

Proof of Theorem fvrn0
StepHypRef Expression
1 id 19 . . 3  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  =  (/) )
2 ssun2 3339 . . . 4  |-  { (/) } 
C_  ( ran  F  u.  { (/) } )
3 0ex 4150 . . . . 5  |-  (/)  e.  _V
43snid 3667 . . . 4  |-  (/)  e.  { (/)
}
52, 4sselii 3177 . . 3  |-  (/)  e.  ( ran  F  u.  { (/)
} )
61, 5syl6eqel 2371 . 2  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  e.  ( ran  F  u.  {
(/) } ) )
7 ssun1 3338 . . 3  |-  ran  F  C_  ( ran  F  u.  {
(/) } )
8 fvprc 5519 . . . . 5  |-  ( -.  X  e.  _V  ->  ( F `  X )  =  (/) )
98con1i 121 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X  e.  _V )
10 fvex 5539 . . . . 5  |-  ( F `
 X )  e. 
_V
1110a1i 10 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  _V )
12 fvbr0 5549 . . . . . 6  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
1312ori 364 . . . . 5  |-  ( -.  X F ( F `
 X )  -> 
( F `  X
)  =  (/) )
1413con1i 121 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X F ( F `  X ) )
15 brelrng 4908 . . . 4  |-  ( ( X  e.  _V  /\  ( F `  X )  e.  _V  /\  X F ( F `  X ) )  -> 
( F `  X
)  e.  ran  F
)
169, 11, 14, 15syl3anc 1182 . . 3  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ran  F )
177, 16sseldi 3178 . 2  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ( ran  F  u.  {
(/) } ) )
186, 17pm2.61i 156 1  |-  ( F `
 X )  e.  ( ran  F  u.  {
(/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640   class class class wbr 4023   ran crn 4690   ` cfv 5255
This theorem is referenced by:  fvssunirn  5551  dfac4  7749  dfac2  7757  dfacacn  7767  axdc2lem  8074  axcclem  8083  ccatfn  11427  plusffval  14379  staffval  15612  scaffval  15645  lpival  15997  ipffval  16552  nmfval  18111  tchex  18649  tchnmfval  18659  orderseqlem  24252  rrnval  26551  lsatset  29180
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-13 1686  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-pow 4188  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-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263
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