ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funfvdm GIF version

Theorem funfvdm 5367
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 5322 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
2 unisng 3670 . . 3 ((𝐹𝐴) ∈ V → {(𝐹𝐴)} = (𝐹𝐴))
31, 2syl 14 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹𝐴))
4 eqid 2088 . . . . 5 dom 𝐹 = dom 𝐹
5 df-fn 5018 . . . . 5 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 887 . . . 4 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7 fnsnfv 5363 . . . 4 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
86, 7sylanbr 279 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
98unieqd 3664 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
103, 9eqtr3d 2122 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  Vcvv 2619  {csn 3446   cuni 3653  dom cdm 4438  cima 4441  Fun wfun 5009   Fn wfn 5010  cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  funfvdm2  5368  fvun1  5370
  Copyright terms: Public domain W3C validator