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Theorem fnfvima 5521
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 5105 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 964 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → Fun 𝐹)
3 simp2 944 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆𝐴)
4 fndm 5107 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
543ad2ant1 964 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → dom 𝐹 = 𝐴)
63, 5sseqtr4d 3063 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆 ⊆ dom 𝐹)
72, 6jca 300 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 simp3 945 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑋𝑆)
9 funfvima2 5519 . 2 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
107, 8, 9sylc 61 1 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  wss 2999  dom cdm 4436  cima 4439  Fun wfun 5004   Fn wfn 5005  cfv 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-fv 5018
This theorem is referenced by:  iseqf1olemnab  9905
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