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Theorem fvrn0 5566
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 3352 . . . 4  |-  { (/) } 
C_  ( ran  F  u.  { (/) } )
3 0ex 4166 . . . . 5  |-  (/)  e.  _V
43snid 3680 . . . 4  |-  (/)  e.  { (/)
}
52, 4sselii 3190 . . 3  |-  (/)  e.  ( ran  F  u.  { (/)
} )
61, 5syl6eqel 2384 . 2  |-  ( ( F `  X )  =  (/)  ->  ( F `
 X )  e.  ( ran  F  u.  {
(/) } ) )
7 ssun1 3351 . . 3  |-  ran  F  C_  ( ran  F  u.  {
(/) } )
8 fvprc 5535 . . . . 5  |-  ( -.  X  e.  _V  ->  ( F `  X )  =  (/) )
98con1i 121 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  X  e.  _V )
10 fvex 5555 . . . . 5  |-  ( F `
 X )  e. 
_V
1110a1i 10 . . . 4  |-  ( -.  ( F `  X
)  =  (/)  ->  ( F `  X )  e.  _V )
12 fvbr0 5565 . . . . . 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 4924 . . . 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 3191 . 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 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653   class class class wbr 4039   ran crn 4706   ` cfv 5271
This theorem is referenced by:  fvssunirn  5567  dfac4  7765  dfac2  7773  dfacacn  7783  axdc2lem  8090  axcclem  8099  ccatfn  11443  plusffval  14395  staffval  15628  scaffval  15661  lpival  16013  ipffval  16568  nmfval  18127  tchex  18665  tchnmfval  18675  orderseqlem  24323  rrnval  26654  lsatset  29802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279
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