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Theorem fnfvima 5756
Description: The function value of an operand in a set is contained in the image of that set, using the  Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
Assertion
Ref Expression
fnfvima  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( F `  X )  e.  ( F " S
) )

Proof of Theorem fnfvima
StepHypRef Expression
1 fnfun 5341 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 976 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  Fun  F )
3 simp2 956 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_  A )
4 fndm 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
543ad2ant1 976 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  dom  F  =  A )
63, 5sseqtr4d 3215 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_ 
dom  F )
72, 6jca 518 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( Fun  F  /\  S  C_  dom  F ) )
8 simp3 957 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  X  e.  S )
9 funfvima2 5754 . 2  |-  ( ( Fun  F  /\  S  C_ 
dom  F )  -> 
( X  e.  S  ->  ( F `  X
)  e.  ( F
" S ) ) )
107, 8, 9sylc 56 1  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( F `  X )  e.  ( F " S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  isomin  5834  isofrlem  5837  fnwelem  6230  php3  7047  fissuni  7160  unxpwdom2  7302  cantnflt  7373  mapfien  7399  dfac12lem2  7770  ackbij2  7869  isf34lem7  8005  isf34lem6  8006  zorn2lem2  8124  ttukeylem5  8140  tskuni  8405  axpre-sup  8791  limsupval2  11954  mhmima  14440  ghmnsgima  14706  dprdfeq0  15257  dprd2dlem1  15276  lmhmima  15804  lmcnp  17032  basqtop  17402  tgqtop  17403  kqfvima  17421  reghmph  17484  uzrest  17592  divstgpopn  17802  divstgplem  17803  cphsqrcl  18620  lhop  19363  ig1peu  19557  ig1pdvds  19562  plypf1  19594  cvmopnlem  23809  nobndlem8  24353  isnumbasgrplem1  27266  psgnunilem1  27416
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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