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Theorem fnfvima 6125
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 5660 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 1015 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  Fun  F )
3 simp2 995 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_  A )
4 fndm 5662 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
543ad2ant1 1015 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  dom  F  =  A )
63, 5sseqtr4d 3526 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_ 
dom  F )
72, 6jca 530 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  ( Fun  F  /\  S  C_  dom  F ) )
8 simp3 996 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  X  e.  S )
9 funfvima2 6123 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   dom cdm 4988   "cima 4991   Fun wfun 5564    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  isomin  6208  isofrlem  6211  fnwelem  6888  php3  7696  fissuni  7817  unxpwdom2  8006  cantnflt  8082  cantnfltOLD  8112  mapfienOLD  8129  dfac12lem2  8515  ackbij2  8614  isf34lem7  8750  isf34lem6  8751  zorn2lem2  8868  ttukeylem5  8884  tskuni  9150  axpre-sup  9535  limsupval2  13385  mhmima  16193  ghmnsgima  16489  psgnunilem1  16717  dprdfeq0  17257  dprdfeq0OLD  17264  dprd2dlem1  17285  lmhmima  17888  lmcnp  19972  basqtop  20378  tgqtop  20379  kqfvima  20397  reghmph  20460  uzrest  20564  qustgpopn  20784  qustgplem  20785  cphsqrtcl  21797  lhop  22583  ig1peu  22738  ig1pdvds  22743  plypf1  22775  f1otrg  24376  fimaproj  28071  txomap  28072  cvmopnlem  28987  mrsubrn  29137  msubrn  29153  nobndlem8  29699  cnambfre  30303  ftc1anclem7  30336  ftc1anc  30338  isnumbasgrplem1  31291  mgmhmima  32862  wfximgfd  38431
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