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Theorem funfv 6996
Description: A simplified expression for the value of a function when we know it is a function. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
funfv (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))

Proof of Theorem funfv
StepHypRef Expression
1 fvex 6919 . . . . 5 (𝐹𝐴) ∈ V
21unisn 4926 . . . 4 {(𝐹𝐴)} = (𝐹𝐴)
3 eqid 2737 . . . . . . 7 dom 𝐹 = dom 𝐹
4 df-fn 6564 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
53, 4mpbiran2 710 . . . . . 6 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6 fnsnfv 6988 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
75, 6sylanbr 582 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
87unieqd 4920 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
92, 8eqtr3id 2791 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
109ex 412 . 2 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴})))
11 ndmfv 6941 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
12 ndmima 6121 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
1312unieqd 4920 . . . 4 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
14 uni0 4935 . . . 4 ∅ = ∅
1513, 14eqtrdi 2793 . . 3 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
1611, 15eqtr4d 2780 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
1710, 16pm2.61d1 180 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  c0 4333  {csn 4626   cuni 4907  dom cdm 5685  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  funfv2  6997  fvun  6999  dffv2  7004  setrecsss  49220
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