MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabss Structured version   Visualization version   GIF version

Theorem opabss 5213
Description: The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
opabss {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
Distinct variable groups:   𝑥,𝑅   𝑦,𝑅

Proof of Theorem opabss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 5212 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)}
2 df-br 5150 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3 eleq1 2813 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
43biimpar 476 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝑧𝑅)
52, 4sylan2b 592 . . . 4 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
65exlimivv 1927 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
76abssi 4063 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)} ⊆ 𝑅
81, 7eqsstri 4011 1 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wss 3944  cop 4636   class class class wbr 5149  {copab 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ss 3961  df-br 5150  df-opab 5212
This theorem is referenced by:  elopabr  5563  elopabran  5564  mptmpoopabbrd  8085  aceq3lem  10145  fullfunc  17898  fthfunc  17899  isfull  17902  isfth  17906  wksv  29505
  Copyright terms: Public domain W3C validator