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Theorem fvrn0 5755
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 21 . . 3  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  =  (/) )
2 ssun2 3513 . . . 4  |-  { (/) } 
C_  ( ran  F  u.  { (/) } )
3 0ex 4341 . . . . 5  |-  (/)  e.  _V
43snid 3843 . . . 4  |-  (/)  e.  { (/)
}
52, 4sselii 3347 . . 3  |-  (/)  e.  ( ran  F  u.  { (/)
} )
61, 5syl6eqel 2526 . 2  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  e.  ( ran  F  u.  {
(/) } ) )
7 ssun1 3512 . . 3  |-  ran  F  C_  ( ran  F  u.  {
(/) } )
8 fvprc 5724 . . . . 5  |-  ( -.  X  e.  _V  ->  ( F `  X )  =  (/) )
98con1i 124 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X  e.  _V )
10 fvex 5744 . . . . 5  |-  ( F `
 X )  e. 
_V
1110a1i 11 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  _V )
12 fvbr0 5754 . . . . . 6  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
1312ori 366 . . . . 5  |-  ( -.  X F ( F `
 X )  -> 
( F `  X
)  =  (/) )
1413con1i 124 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X F ( F `  X ) )
15 brelrng 5101 . . . 4  |-  ( ( X  e.  _V  /\  ( F `  X )  e.  _V  /\  X F ( F `  X ) )  -> 
( F `  X
)  e.  ran  F
)
169, 11, 14, 15syl3anc 1185 . . 3  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ran  F )
177, 16sseldi 3348 . 2  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ( ran  F  u.  {
(/) } ) )
186, 17pm2.61i 159 1  |-  ( F `
 X )  e.  ( ran  F  u.  {
(/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320   (/)c0 3630   {csn 3816   class class class wbr 4214   ran crn 4881   ` cfv 5456
This theorem is referenced by:  fvssunirn  5756  dfac4  8005  dfac2  8013  dfacacn  8023  axdc2lem  8330  axcclem  8339  ccatfn  11743  plusffval  14704  staffval  15937  scaffval  15970  lpival  16318  ipffval  16881  nmfval  18638  tchex  19178  tchnmfval  19188  orderseqlem  25529  rrnval  26538  lsatset  29850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891  df-iota 5420  df-fv 5464
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