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Theorem fvrn0 5879
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 3661 . . . 4  |-  { (/) } 
C_  ( ran  F  u.  { (/) } )
3 0ex 4570 . . . . 5  |-  (/)  e.  _V
43snid 4048 . . . 4  |-  (/)  e.  { (/)
}
52, 4sselii 3494 . . 3  |-  (/)  e.  ( ran  F  u.  { (/)
} )
61, 5syl6eqel 2556 . 2  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  e.  ( ran  F  u.  {
(/) } ) )
7 ssun1 3660 . . 3  |-  ran  F  C_  ( ran  F  u.  {
(/) } )
8 fvprc 5851 . . . . 5  |-  ( -.  X  e.  _V  ->  ( F `  X )  =  (/) )
98con1i 129 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X  e.  _V )
10 fvex 5867 . . . . 5  |-  ( F `
 X )  e. 
_V
1110a1i 11 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  _V )
12 fvbr0 5878 . . . . . 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 5223 . . . 4  |-  ( ( X  e.  _V  /\  ( F `  X )  e.  _V  /\  X F ( F `  X ) )  -> 
( F `  X
)  e.  ran  F
)
169, 11, 14, 15syl3anc 1223 . . 3  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  ran  F )
177, 16sseldi 3495 . 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 1374    e. wcel 1762   _Vcvv 3106    u. cun 3467   (/)c0 3778   {csn 4020   class class class wbr 4440   ran crn 4993   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-cnv 5000  df-dm 5002  df-rn 5003  df-iota 5542  df-fv 5587
This theorem is referenced by:  fvssunirn  5880  dfac4  8492  dfac2  8500  dfacacn  8510  axdc2lem  8817  axcclem  8826  ccatfn  12543  plusffval  15733  staffval  17272  scaffval  17306  lpival  17668  ipffval  18443  nmfval  20837  tchex  21388  tchnmfval  21399  orderseqlem  28895  rrnval  29913  lsatset  33662
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