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

Theorem funfvdm 5157
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 𝐹 A dom 𝐹) → (𝐹A) = (𝐹 “ {A}))

Proof of Theorem funfvdm
StepHypRef Expression
1 funfvex 5113 . . 3 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
2 unisng 3567 . . 3 ((𝐹A) V → {(𝐹A)} = (𝐹A))
31, 2syl 14 . 2 ((Fun 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹A))
4 eqid 2018 . . . . 5 dom 𝐹 = dom 𝐹
5 df-fn 4828 . . . . 5 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 dom 𝐹 = dom 𝐹))
64, 5mpbiran2 834 . . . 4 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7 fnsnfv 5153 . . . 4 ((𝐹 Fn dom 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹 “ {A}))
86, 7sylanbr 269 . . 3 ((Fun 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹 “ {A}))
98unieqd 3561 . 2 ((Fun 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹 “ {A}))
103, 9eqtr3d 2052 1 ((Fun 𝐹 A dom 𝐹) → (𝐹A) = (𝐹 “ {A}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  Vcvv 2531  {csn 3346   cuni 3550  dom cdm 4268  cima 4271  Fun wfun 4819   Fn wfn 4820  cfv 4825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by:  funfvdm2  5158  fvun1  5160
  Copyright terms: Public domain W3C validator