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Theorem ffvelcdmi 5651
Description: A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
Hypothesis
Ref Expression
ffvelcdmi.1  |-  F : A
--> B
Assertion
Ref Expression
ffvelcdmi  |-  ( C  e.  A  ->  ( F `  C )  e.  B )

Proof of Theorem ffvelcdmi
StepHypRef Expression
1 ffvelcdmi.1 . 2  |-  F : A
--> B
2 ffvelcdm 5650 . 2  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F `  C
)  e.  B )
31, 2mpan 424 1  |-  ( C  e.  A  ->  ( F `  C )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   -->wf 5213   ` cfv 5217
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225
This theorem is referenced by:  omgadd  10782  cjcl  10857  climmpt  11308  cn1lem  11322  climcn1lem  11327  fsumrelem  11479  efcl  11672  sincl  11714  coscl  11715  algcvg  12048  algcvgb  12050  algcvga  12051  algfx  12052  eucalgcvga  12058  eucalg  12059  sqpweven  12175  2sqpwodd  12176  ennnfonelemnn0  12423  relogcl  14286  nninfomnilem  14770
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