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Theorem funfvdm 5600
Description: A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
Assertion
Ref Expression
funfvdm ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))

Proof of Theorem funfvdm
StepHypRef Expression
1 funfvex 5551 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
2 unisng 3841 . . 3 ((𝐹𝐴) ∈ V → {(𝐹𝐴)} = (𝐹𝐴))
31, 2syl 14 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹𝐴))
4 eqid 2189 . . . . 5 dom 𝐹 = dom 𝐹
5 df-fn 5238 . . . . 5 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 943 . . . 4 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7 fnsnfv 5596 . . . 4 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
86, 7sylanbr 285 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
98unieqd 3835 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
103, 9eqtr3d 2224 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3607   cuni 3824  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:  funfvdm2  5601  fvun1  5603
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