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Theorem fnfvelrn 5612
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 5611 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  ran  F
)
21funfni 5283 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F `  B
)  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2135   ran crn 4600    Fn wfn 5178   ` cfv 5183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-sbc 2948  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-iota 5148  df-fun 5185  df-fn 5186  df-fv 5191
This theorem is referenced by:  ffvelrn  5613  fnovrn  5981  fo1stresm  6122  fo2ndresm  6123  fo2ndf  6187  phplem4  6813  phplem4on  6825  cc2lem  7199  frec2uzrand  10331  frecuzrdglem  10337  frecuzrdg0  10339  frecuzrdg0t  10348  uzin2  10919
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