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Theorem funfv 6304
Description: A simplified expression for the value of a function when we know it's 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 6239 . . . . 5 (𝐹𝐴) ∈ V
21unisn 4483 . . . 4 {(𝐹𝐴)} = (𝐹𝐴)
3 eqid 2651 . . . . . . 7 dom 𝐹 = dom 𝐹
4 df-fn 5929 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
53, 4mpbiran2 974 . . . . . 6 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6 fnsnfv 6297 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
75, 6sylanbr 489 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
87unieqd 4478 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
92, 8syl5eqr 2699 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
109ex 449 . 2 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴})))
11 ndmfv 6256 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
12 ndmima 5537 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
1312unieqd 4478 . . . 4 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
14 uni0 4497 . . . 4 ∅ = ∅
1513, 14syl6eq 2701 . . 3 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
1611, 15eqtr4d 2688 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
1710, 16pm2.61d1 171 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  c0 3948  {csn 4210   cuni 4468  dom cdm 5143  cima 5146  Fun wfun 5920   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  funfv2  6305  fvun  6307  dffv2  6310  setrecsss  42772
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