|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > fv2 | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| fv2 | ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-fv 6568 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
| 2 | dfiota2 6514 | . 2 ⊢ (℩𝑦𝐴𝐹𝑦) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)} | |
| 3 | 1, 2 | eqtri 2764 | 1 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 = wceq 1539 {cab 2713 ∪ cuni 4906 class class class wbr 5142 ℩cio 6511 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-sn 4626 df-uni 4907 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: elfv 6903 | 
| Copyright terms: Public domain | W3C validator |