Theorem fv2 5549

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	⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fv2

Step	Hyp	Ref	Expression
1		df-fv 5262	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		dfiota2 5216	. 2 ⊢ (℩𝑦𝐴𝐹𝑦) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)}
3	1, 2	eqtri 2214	1 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)}

Syntax hints:   ↔ wb 105  ∀wal 1362   = wceq 1364  {cab 2179  ∪ cuni 3835   class class class wbr 4029  ℩cio 5213  ‘cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-sn 3624  df-uni 3836  df-iota 5215  df-fv 5262

This theorem is referenced by:  elfv  5552