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Theorem fnfvima 5772
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 5332 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1020 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → Fun 𝐹)
3 simp2 1000 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆𝐴)
4 fndm 5334 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
543ad2ant1 1020 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → dom 𝐹 = 𝐴)
63, 5sseqtrrd 3209 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆 ⊆ dom 𝐹)
72, 6jca 306 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 simp3 1001 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑋𝑆)
9 funfvima2 5770 . 2 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
107, 8, 9sylc 62 1 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wss 3144  dom cdm 4644  cima 4647  Fun wfun 5229   Fn wfn 5230  cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  iseqf1olemnab  10519  ennnfonelemrn  12470  mhmima  12943  ghmnsgima  13207  lmtopcnp  14207
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