 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fv2 Structured version   Visualization version   GIF version

Theorem fv2 6224
 Description: Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fv2 (𝐹𝐴) = {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥)}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fv2
StepHypRef Expression
1 df-fv 5934 . 2 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 dfiota2 5890 . 2 (℩𝑦𝐴𝐹𝑦) = {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥)}
31, 2eqtri 2673 1 (𝐹𝐴) = {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wal 1521   = wceq 1523  {cab 2637  ∪ cuni 4468   class class class wbr 4685  ℩cio 5887  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-sn 4211  df-uni 4469  df-iota 5889  df-fv 5934 This theorem is referenced by:  elfv  6227
 Copyright terms: Public domain W3C validator