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Theorem fveu 6640
 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 6336 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotauni 6303 . 2 (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = {𝑥𝐴𝐹𝑥})
31, 2syl5eq 2848 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∃!weu 2631  {cab 2779  ∪ cuni 4803   class class class wbr 5033  ℩cio 6285  ‘cfv 6328 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-uni 4804  df-iota 6287  df-fv 6336 This theorem is referenced by:  afveu  43706
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