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Theorem fafvelrn 28138
Description: A function's value belongs to its codomain, analogous to ffvelrn 5679. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fafvelrn  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B
)

Proof of Theorem fafvelrn
StepHypRef Expression
1 ffn 5405 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fnafvelrn 28137 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
31, 2sylan 457 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  ran  F )
4 frn 5411 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
54sseld 3192 . . 3  |-  ( F : A --> B  -> 
( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
65adantr 451 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( ( F''' C )  e.  ran  F  -> 
( F''' C )  e.  B
) )
73, 6mpd 14 1  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   ran crn 4706    Fn wfn 5266   -->wf 5267  '''cafv 28075
This theorem is referenced by:  ffnafv  28139
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-dfat 28077  df-afv 28078
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