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Theorem fnfvima 6493
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 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))

Proof of Theorem fnfvima
StepHypRef Expression
1 fnfun 5986 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1081 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → Fun 𝐹)
3 simp2 1061 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆𝐴)
4 fndm 5988 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
543ad2ant1 1081 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → dom 𝐹 = 𝐴)
63, 5sseqtr4d 3640 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆 ⊆ dom 𝐹)
72, 6jca 554 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 simp3 1062 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑋𝑆)
9 funfvima2 6490 . 2 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
107, 8, 9sylc 65 1 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wss 3572  dom cdm 5112  cima 5115  Fun wfun 5880   Fn wfn 5881  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894
This theorem is referenced by:  isomin  6584  isofrlem  6587  fnwelem  7289  php3  8143  fissuni  8268  unxpwdom2  8490  cantnflt  8566  dfac12lem2  8963  ackbij2  9062  isf34lem7  9198  isf34lem6  9199  zorn2lem2  9316  ttukeylem5  9332  tskuni  9602  axpre-sup  9987  limsupval2  14205  mhmima  17357  ghmnsgima  17678  psgnunilem1  17907  dprdfeq0  18415  dprd2dlem1  18434  lmhmima  19041  lmcnp  21102  basqtop  21508  tgqtop  21509  kqfvima  21527  reghmph  21590  uzrest  21695  qustgpopn  21917  qustgplem  21918  cphsqrtcl  22978  lhop  23773  ig1peu  23925  ig1pdvds  23930  plypf1  23962  f1otrg  25745  fimaproj  29885  txomap  29886  sitgaddlemb  30395  cvmopnlem  31245  mrsubrn  31395  msubrn  31411  nosupno  31833  nosupbday  31835  noetalem3  31849  scutun12  31901  scutbdaybnd  31905  scutbdaylt  31906  poimirlem4  33393  poimirlem6  33395  poimirlem7  33396  poimirlem16  33405  poimirlem17  33406  poimirlem19  33408  poimirlem20  33409  poimirlem23  33412  cnambfre  33438  ftc1anclem7  33471  ftc1anc  33473  isnumbasgrplem1  37497  wfximgfd  38289  funimaeq  39283  fnfvima2  39300  mgmhmima  41573
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