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Theorem funfvima2 5966
Description: A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
Assertion
Ref Expression
funfvima2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )

Proof of Theorem funfvima2
StepHypRef Expression
1 ssel 3334 . . 3  |-  ( A 
C_  dom  F  ->  ( B  e.  A  ->  B  e.  dom  F ) )
2 funfvima 5965 . . . . . 6  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
32ex 424 . . . . 5  |-  ( Fun 
F  ->  ( B  e.  dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
43com23 74 . . . 4  |-  ( Fun 
F  ->  ( B  e.  A  ->  ( B  e.  dom  F  -> 
( F `  B
)  e.  ( F
" A ) ) ) )
54a2d 24 . . 3  |-  ( Fun 
F  ->  ( ( B  e.  A  ->  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
61, 5syl5 30 . 2  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A
) ) ) )
76imp 419 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    C_ wss 3312   dom cdm 4870   "cima 4873   Fun wfun 5440   ` cfv 5446
This theorem is referenced by:  fnfvima  5968  f1oweALT  6066  tz7.49  6694  phimullem  13160  mrcuni  13838  iscldtop  17151  1stcfb  17500  2ndcomap  17513  rnelfm  17977  fmfnfmlem2  17979  fmfnfmlem4  17981  qtopbaslem  18784  tgqioo  18823  bndth  18975  volsup  19442  dyadmbllem  19483  opnmbllem  19485  itg1addlem4  19583  c1liplem1  19872  dvcnvrelem1  19893  dvcnvrelem2  19894  plyco0  20103  plyaddlem1  20124  plymullem1  20125  dvloglem  20531  logf1o2  20533  efopn  20541  eupares  21689  imaelshi  23553  funimass4f  24036  sitgclg  24648  cvmliftlem3  24966  nocvxminlem  25637  nocvxmin  25638  axcontlem10  25904  mblfinlem  26234  ivthALT  26329  ismtyres  26508  heibor1lem  26509  ismrc  26746  aomclem4  27123  frlmsslsp  27216  lindfrn  27259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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