ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funfvdm2 GIF version

Theorem funfvdm2 5403
Description: The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
Assertion
Ref Expression
funfvdm2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funfvdm2
StepHypRef Expression
1 funfvdm 5402 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
2 imasng 4830 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
32adantl 272 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
43unieqd 3686 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
51, 4eqtrd 2127 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  {cab 2081  {csn 3466   cuni 3675   class class class wbr 3867  dom cdm 4467  cima 4470  Fun wfun 5043  cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057
This theorem is referenced by:  funfvdm2f  5404
  Copyright terms: Public domain W3C validator