Theorem fveu 6856

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 6529	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotauni 6498	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
3	1, 2	eqtrid 2809	1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Syntax hints:   → wi 4   = wceq 1560  ∃!weu 2595  {cab 2740  ∪ cuni 4865   class class class wbr 5100  ℩cio 6475  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-sn 4583  df-pr 4585  df-uni 4866  df-iota 6477  df-fv 6529

This theorem is referenced by:  afveu  47747