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Theorem fnfvelrn 5669
Description: A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
Assertion
Ref Expression
fnfvelrn  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F `  B
)  e.  ran  F
)

Proof of Theorem fnfvelrn
StepHypRef Expression
1 fvelrn 5668 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  ran  F
)
21funfni 5335 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F `  B
)  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   ran crn 4645    Fn wfn 5230   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  ffvelcdm  5670  fnovrn  6044  fo1stresm  6186  fo2ndresm  6187  fo2ndf  6252  phplem4  6883  phplem4on  6895  cc2lem  7295  frec2uzrand  10436  frecuzrdglem  10442  frecuzrdg0  10444  frecuzrdg0t  10453  uzin2  11028  ghmrn  13196  conjnmz  13218
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