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

Theorem fveu 6828
Description: The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
fveu (∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem fveu
StepHypRef Expression
1 df-fv 6501 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotauni 6468 . 2 (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = {𝑥𝐴𝐹𝑥})
31, 2eqtrid 2789 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  ∃!weu 2567  {cab 2714   cuni 4863   class class class wbr 5103  cio 6443  cfv 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3445  df-un 3913  df-in 3915  df-ss 3925  df-sn 4585  df-pr 4587  df-uni 4864  df-iota 6445  df-fv 6501
This theorem is referenced by:  afveu  45286
  Copyright terms: Public domain W3C validator