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Theorem fnfvima 6136
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 5676 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 1017 . . 3  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  Fun  F )
3 simp2 997 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  S  C_  A )
4 fndm 5678 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
543ad2ant1 1017 . . . 4  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  dom  F  =  A )
63, 5sseqtr4d 3541 . . 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 998 . 2  |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S )  ->  X  e.  S )
9 funfvima2 6134 . 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 973    = wceq 1379    e. wcel 1767    C_ wss 3476   dom cdm 4999   "cima 5002   Fun wfun 5580    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  isomin  6219  isofrlem  6222  fnwelem  6895  php3  7700  fissuni  7821  unxpwdom2  8010  cantnflt  8087  cantnfltOLD  8117  mapfienOLD  8134  dfac12lem2  8520  ackbij2  8619  isf34lem7  8755  isf34lem6  8756  zorn2lem2  8873  ttukeylem5  8889  tskuni  9157  axpre-sup  9542  limsupval2  13262  mhmima  15804  ghmnsgima  16085  psgnunilem1  16314  dprdfeq0  16852  dprdfeq0OLD  16859  dprd2dlem1  16880  lmhmima  17476  lmcnp  19571  basqtop  19947  tgqtop  19948  kqfvima  19966  reghmph  20029  uzrest  20133  divstgpopn  20353  divstgplem  20354  cphsqrtcl  21366  lhop  22152  ig1peu  22307  ig1pdvds  22312  plypf1  22344  f1otrg  23850  fimaproj  27499  txomap  27500  cvmopnlem  28363  nobndlem8  29036  cnambfre  29640  ftc1anclem7  29673  ftc1anc  29675  isnumbasgrplem1  30654
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