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Theorem fnfvima 5943
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 5509 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 978 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  Fun  F )
3 simp2 958 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_  A )
4 fndm 5511 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
543ad2ant1 978 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  dom  F  =  A )
63, 5sseqtr4d 3353 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_ 
dom  F )
72, 6jca 519 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( Fun  F  /\  S  C_  dom  F ) )
8 simp3 959 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  X  e.  S )
9 funfvima2 5941 . 2  |-  ( ( Fun  F  /\  S  C_ 
dom  F )  -> 
( X  e.  S  ->  ( F `  X
)  e.  ( F
" S ) ) )
107, 8, 9sylc 58 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3288   dom cdm 4845   "cima 4848   Fun wfun 5415    Fn wfn 5416   ` cfv 5421
This theorem is referenced by:  isomin  6024  isofrlem  6027  fnwelem  6428  php3  7260  fissuni  7377  unxpwdom2  7520  cantnflt  7591  mapfien  7617  dfac12lem2  7988  ackbij2  8087  isf34lem7  8223  isf34lem6  8224  zorn2lem2  8341  ttukeylem5  8357  tskuni  8622  axpre-sup  9008  limsupval2  12237  mhmima  14726  ghmnsgima  14992  dprdfeq0  15543  dprd2dlem1  15562  lmhmima  16086  lmcnp  17330  basqtop  17704  tgqtop  17705  kqfvima  17723  reghmph  17786  uzrest  17890  divstgpopn  18110  divstgplem  18111  cphsqrcl  19108  lhop  19861  ig1peu  20055  ig1pdvds  20060  plypf1  20092  cvmopnlem  24926  nobndlem8  25575  cnambfre  26162  isnumbasgrplem1  27142  psgnunilem1  27292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429
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