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

Theorem opabssxp 4837
 Description: An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
Assertion
Ref Expression
opabssxp {x, y ((x A y B) φ)} (A × B)
Distinct variable groups:   x,y,A   x,B,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem opabssxp
StepHypRef Expression
1 simpl 443 . . 3 (((x A y B) φ) → (x A y B))
21ssopab2i 4714 . 2 {x, y ((x A y B) φ)} {x, y (x A y B)}
3 df-xp 4784 . 2 (A × B) = {x, y (x A y B)}
42, 3sseqtr4i 3304 1 {x, y ((x A y B) φ)} (A × B)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∈ wcel 1710   ⊆ wss 3257  {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-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-xp 4784 This theorem is referenced by:  dmoprabss  5575
 Copyright terms: Public domain W3C validator