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Theorem fnfvima 6374
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 5884 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1074 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → Fun 𝐹)
3 simp2 1054 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆𝐴)
4 fndm 5886 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
543ad2ant1 1074 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → dom 𝐹 = 𝐴)
63, 5sseqtr4d 3600 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆 ⊆ dom 𝐹)
72, 6jca 552 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 simp3 1055 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑋𝑆)
9 funfvima2 6371 . 2 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
107, 8, 9sylc 62 1 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wss 3535  dom cdm 5024  cima 5027  Fun wfun 5780   Fn wfn 5781  cfv 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-fv 5794
This theorem is referenced by:  isomin  6461  isofrlem  6464  fnwelem  7152  php3  8004  fissuni  8127  unxpwdom2  8349  cantnflt  8425  dfac12lem2  8822  ackbij2  8921  isf34lem7  9057  isf34lem6  9058  zorn2lem2  9175  ttukeylem5  9191  tskuni  9457  axpre-sup  9842  limsupval2  14001  mhmima  17128  ghmnsgima  17449  psgnunilem1  17678  dprdfeq0  18186  dprd2dlem1  18205  lmhmima  18810  lmcnp  20856  basqtop  21262  tgqtop  21263  kqfvima  21281  reghmph  21344  uzrest  21449  qustgpopn  21671  qustgplem  21672  cphsqrtcl  22712  lhop  23496  ig1peu  23648  ig1pdvds  23653  plypf1  23685  f1otrg  25465  fimaproj  29030  txomap  29031  sitgaddlemb  29539  cvmopnlem  30316  mrsubrn  30466  msubrn  30482  nobndlem8  30900  poimirlem4  32382  poimirlem6  32384  poimirlem7  32385  poimirlem16  32394  poimirlem17  32395  poimirlem19  32397  poimirlem20  32398  poimirlem23  32401  cnambfre  32427  ftc1anclem7  32460  ftc1anc  32462  isnumbasgrplem1  36489  wfximgfd  37284  mgmhmima  41590
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