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Theorem fnfvima 5958
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 5511 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 1009 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  Fun  F )
3 simp2 989 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_  A )
4 fndm 5513 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
543ad2ant1 1009 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  dom  F  =  A )
63, 5sseqtr4d 3396 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_ 
dom  F )
72, 6jca 532 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( Fun  F  /\  S  C_  dom  F ) )
8 simp3 990 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  X  e.  S )
9 funfvima2 5956 . 2  |-  ( ( Fun  F  /\  S  C_ 
dom  F )  -> 
( X  e.  S  ->  ( F `  X
)  e.  ( F
" S ) ) )
107, 8, 9sylc 60 1  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( F `  X )  e.  ( F " S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3331   dom cdm 4843   "cima 4846   Fun wfun 5415    Fn wfn 5416   ` cfv 5421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pr 4534
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-fv 5429
This theorem is referenced by:  isomin  6031  isofrlem  6034  fnwelem  6690  php3  7500  fissuni  7619  unxpwdom2  7806  cantnflt  7883  cantnfltOLD  7913  mapfienOLD  7930  dfac12lem2  8316  ackbij2  8415  isf34lem7  8551  isf34lem6  8552  zorn2lem2  8669  ttukeylem5  8685  tskuni  8953  axpre-sup  9339  limsupval2  12961  mhmima  15494  ghmnsgima  15773  psgnunilem1  16002  dprdfeq0  16515  dprdfeq0OLD  16522  dprd2dlem1  16543  lmhmima  17131  lmcnp  18911  basqtop  19287  tgqtop  19288  kqfvima  19306  reghmph  19369  uzrest  19473  divstgpopn  19693  divstgplem  19694  cphsqrcl  20706  lhop  21491  ig1peu  21646  ig1pdvds  21651  plypf1  21683  f1otrg  23120  cvmopnlem  27170  nobndlem8  27843  cnambfre  28443  ftc1anclem7  28476  ftc1anc  28478  isnumbasgrplem1  29460
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