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Theorem funfv 6749
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 6682 . . . . 5 (𝐹𝐴) ∈ V
21unisn 4853 . . . 4 {(𝐹𝐴)} = (𝐹𝐴)
3 eqid 2826 . . . . . . 7 dom 𝐹 = dom 𝐹
4 df-fn 6357 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
53, 4mpbiran2 706 . . . . . 6 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6 fnsnfv 6742 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
75, 6sylanbr 582 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
87unieqd 4847 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
92, 8syl5eqr 2875 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
109ex 413 . 2 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴})))
11 ndmfv 6699 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
12 ndmima 5965 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
1312unieqd 4847 . . . 4 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
14 uni0 4864 . . . 4 ∅ = ∅
1513, 14syl6eq 2877 . . 3 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
1611, 15eqtr4d 2864 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
1710, 16pm2.61d1 181 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  c0 4295  {csn 4564   cuni 4837  dom cdm 5554  cima 5557  Fun wfun 6348   Fn wfn 6349  cfv 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-fv 6362
This theorem is referenced by:  funfv2  6750  fvun  6752  dffv2  6755  setrecsss  44705
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