Theorem fveu 6829

Description: The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)

Assertion

Ref	Expression
fveu	⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem fveu

Step	Hyp	Ref	Expression
1		df-fv 6506	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotauni 6475	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
3	1, 2	eqtrid 2783	1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Syntax hints:   → wi 4   = wceq 1542  ∃!weu 2568  {cab 2714  ∪ cuni 4850   class class class wbr 5085  ℩cio 6452  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-sn 4568  df-pr 4570  df-uni 4851  df-iota 6454  df-fv 6506

This theorem is referenced by:  afveu  47601