Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funfv Structured version   Visualization version   GIF version

Theorem funfv 6743
 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 6675 . . . . 5 (𝐹𝐴) ∈ V
21unisn 4823 . . . 4 {(𝐹𝐴)} = (𝐹𝐴)
3 eqid 2758 . . . . . . 7 dom 𝐹 = dom 𝐹
4 df-fn 6342 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
53, 4mpbiran2 709 . . . . . 6 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6 fnsnfv 6735 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
75, 6sylanbr 585 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
87unieqd 4815 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
92, 8syl5eqr 2807 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
109ex 416 . 2 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴})))
11 ndmfv 6692 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
12 ndmima 5942 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
1312unieqd 4815 . . . 4 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
14 uni0 4831 . . . 4 ∅ = ∅
1513, 14eqtrdi 2809 . . 3 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
1611, 15eqtr4d 2796 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
1710, 16pm2.61d1 183 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4227  {csn 4525  ∪ cuni 4801  dom cdm 5527   “ cima 5530  Fun wfun 6333   Fn wfn 6334  ‘cfv 6339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-fv 6347 This theorem is referenced by:  funfv2  6744  fvun  6746  dffv2  6751  setrecsss  45694
 Copyright terms: Public domain W3C validator