New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  fconstopab GIF version

Theorem fconstopab 4815
 Description: Representation of a constant function using ordered pairs. (Contributed by NM, 12-Oct-1999.)
Assertion
Ref Expression
fconstopab (A × {B}) = {x, y (x A y = B)}
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem fconstopab
StepHypRef Expression
1 df-xp 4784 . 2 (A × {B}) = {x, y (x A y {B})}
2 df-sn 3741 . . . . 5 {B} = {y y = B}
32abeq2i 2460 . . . 4 (y {B} ↔ y = B)
43anbi2i 675 . . 3 ((x A y {B}) ↔ (x A y = B))
54opabbii 4626 . 2 {x, y (x A y {B})} = {x, y (x A y = B)}
61, 5eqtri 2373 1 (A × {B}) = {x, y (x A y = B)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {csn 3737  {copab 4622   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sn 3741  df-opab 4623  df-xp 4784 This theorem is referenced by:  fconst  5250  fopabsn  5441  fconstmpt  5681
 Copyright terms: Public domain W3C validator