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Theorem ffvelcdmi 5646
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 5645 . 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 5208   ` cfv 5212
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 4118  ax-pow 4171  ax-pr 4206
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 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220
This theorem is referenced by:  omgadd  10763  cjcl  10838  climmpt  11289  cn1lem  11303  climcn1lem  11308  fsumrelem  11460  efcl  11653  sincl  11695  coscl  11696  algcvg  12028  algcvgb  12030  algcvga  12031  algfx  12032  eucalgcvga  12038  eucalg  12039  sqpweven  12155  2sqpwodd  12156  ennnfonelemnn0  12403  relogcl  13943  nninfomnilem  14416
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