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Theorem funfvima2 5754
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 3174 . . 3  |-  ( A 
C_  dom  F  ->  ( B  e.  A  ->  B  e.  dom  F ) )
2 funfvima 5753 . . . . . 6  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
32ex 423 . . . . 5  |-  ( Fun 
F  ->  ( B  e.  dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
43com23 72 . . . 4  |-  ( Fun 
F  ->  ( B  e.  A  ->  ( B  e.  dom  F  -> 
( F `  B
)  e.  ( F
" A ) ) ) )
54a2d 23 . . 3  |-  ( Fun 
F  ->  ( ( B  e.  A  ->  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
61, 5syl5 28 . 2  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A
) ) ) )
76imp 418 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 358    e. wcel 1684    C_ wss 3152   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  fnfvima  5756  f1oweALT  5851  tz7.49  6457  phimullem  12847  mrcuni  13523  iscldtop  16832  1stcfb  17171  2ndcomap  17184  rnelfm  17648  fmfnfmlem2  17650  fmfnfmlem4  17652  qtopbaslem  18267  tgqioo  18306  bndth  18456  volsup  18913  dyadmbllem  18954  opnmbllem  18956  itg1addlem4  19054  c1liplem1  19343  dvcnvrelem1  19364  dvcnvrelem2  19365  plyco0  19574  plyaddlem1  19595  plymullem1  19596  dvloglem  19995  logf1o2  19997  efopn  20005  imaelshi  22638  funimass4f  23042  cvmliftlem3  23818  eupares  23899  nocvxminlem  24344  nocvxmin  24345  axcontlem10  24601  lvsovso  25626  ivthALT  26258  ismtyres  26532  heibor1lem  26533  ismrc  26776  aomclem4  27154  frlmsslsp  27248  lindfrn  27291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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