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

Theorem funfvdm2 5162
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 𝐹 A dom 𝐹) → (𝐹A) = {yA𝐹y})
Distinct variable groups:   y,A   y,𝐹

Proof of Theorem funfvdm2
StepHypRef Expression
1 funfvdm 5161 . 2 ((Fun 𝐹 A dom 𝐹) → (𝐹A) = (𝐹 “ {A}))
2 imasng 4617 . . . 4 (A dom 𝐹 → (𝐹 “ {A}) = {yA𝐹y})
32adantl 262 . . 3 ((Fun 𝐹 A dom 𝐹) → (𝐹 “ {A}) = {yA𝐹y})
43unieqd 3565 . 2 ((Fun 𝐹 A dom 𝐹) → (𝐹 “ {A}) = {yA𝐹y})
51, 4eqtrd 2054 1 ((Fun 𝐹 A dom 𝐹) → (𝐹A) = {yA𝐹y})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  {cab 2008  {csn 3350   cuni 3554   class class class wbr 3738  dom cdm 4272  cima 4275  Fun wfun 4823  cfv 4829
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  funfvdm2f  5163
  Copyright terms: Public domain W3C validator