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Theorem fnfvima 6536
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 6026 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1102 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → Fun 𝐹)
3 simp2 1082 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆𝐴)
4 fndm 6028 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
543ad2ant1 1102 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → dom 𝐹 = 𝐴)
63, 5sseqtr4d 3675 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆 ⊆ dom 𝐹)
72, 6jca 553 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 simp3 1083 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑋𝑆)
9 funfvima2 6533 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  dom cdm 5143  cima 5146  Fun wfun 5920   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  isomin  6627  isofrlem  6630  fnwelem  7337  php3  8187  fissuni  8312  unxpwdom2  8534  cantnflt  8607  dfac12lem2  9004  ackbij2  9103  isf34lem7  9239  isf34lem6  9240  zorn2lem2  9357  ttukeylem5  9373  tskuni  9643  axpre-sup  10028  limsupval2  14255  mhmima  17410  ghmnsgima  17731  psgnunilem1  17959  dprdfeq0  18467  dprd2dlem1  18486  lmhmima  19095  lmcnp  21156  basqtop  21562  tgqtop  21563  kqfvima  21581  reghmph  21644  uzrest  21748  qustgpopn  21970  qustgplem  21971  cphsqrtcl  23030  lhop  23824  ig1peu  23976  ig1pdvds  23981  plypf1  24013  f1otrg  25796  fimaproj  30028  txomap  30029  sitgaddlemb  30538  cvmopnlem  31386  mrsubrn  31536  msubrn  31552  nosupno  31974  nosupbday  31976  noetalem3  31990  scutun12  32042  scutbdaybnd  32046  scutbdaylt  32047  poimirlem4  33543  poimirlem6  33545  poimirlem7  33546  poimirlem16  33555  poimirlem17  33556  poimirlem19  33558  poimirlem20  33559  poimirlem23  33562  cnambfre  33588  ftc1anclem7  33621  ftc1anc  33623  isnumbasgrplem1  37988  wfximgfd  38780  funimaeq  39775  fnfvima2  39792  mgmhmima  42127
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