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Theorem opabss 5116
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 5115 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)}
2 df-br 5053 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3 eleq1 2903 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
43biimpar 481 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝑧𝑅)
52, 4sylan2b 596 . . . 4 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
65exlimivv 1934 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
76abssi 4031 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)} ⊆ 𝑅
81, 7eqsstri 3986 1 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wss 3919  cop 4555   class class class wbr 5052  {copab 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-in 3926  df-ss 3936  df-br 5053  df-opab 5115
This theorem is referenced by:  aceq3lem  9538  fullfunc  17172  fthfunc  17173  isfull  17176  isfth  17180
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