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Theorem fvrn0 5822
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 22 . . 3  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  =  (/) )
2 ssun2 3629 . . . 4  |-  { (/) } 
C_  ( ran  F  u.  { (/) } )
3 0ex 4531 . . . . 5  |-  (/)  e.  _V
43snid 4014 . . . 4  |-  (/)  e.  { (/)
}
52, 4sselii 3462 . . 3  |-  (/)  e.  ( ran  F  u.  { (/)
} )
61, 5syl6eqel 2550 . 2  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  e.  ( ran  F  u.  {
(/) } ) )
7 ssun1 3628 . . 3  |-  ran  F  C_  ( ran  F  u.  {
(/) } )
8 fvprc 5794 . . . . 5  |-  ( -.  X  e.  _V  ->  ( F `  X )  =  (/) )
98con1i 129 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X  e.  _V )
10 fvex 5810 . . . . 5  |-  ( F `
 X )  e. 
_V
1110a1i 11 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  _V )
12 fvbr0 5821 . . . . . 6  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
1312ori 375 . . . . 5  |-  ( -.  X F ( F `
 X )  -> 
( F `  X
)  =  (/) )
1413con1i 129 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X F ( F `  X ) )
15 brelrng 5178 . . . 4  |-  ( ( X  e.  _V  /\  ( F `  X )  e.  _V  /\  X F ( F `  X ) )  -> 
( F `  X
)  e.  ran  F
)
169, 11, 14, 15syl3anc 1219 . . 3  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ran  F )
177, 16sseldi 3463 . 2  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ( ran  F  u.  {
(/) } ) )
186, 17pm2.61i 164 1  |-  ( F `
 X )  e.  ( ran  F  u.  {
(/) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3435   (/)c0 3746   {csn 3986   class class class wbr 4401   ran crn 4950   ` cfv 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-cnv 4957  df-dm 4959  df-rn 4960  df-iota 5490  df-fv 5535
This theorem is referenced by:  fvssunirn  5823  dfac4  8404  dfac2  8412  dfacacn  8422  axdc2lem  8729  axcclem  8738  ccatfn  12391  plusffval  15547  staffval  17056  scaffval  17090  lpival  17451  ipffval  18203  nmfval  20314  tchex  20865  tchnmfval  20876  orderseqlem  27858  rrnval  28875  lsatset  32974
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