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Theorem opabss 4028
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 4026 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)}
2 df-br 3966 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3 eleq1 2220 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
43biimpar 295 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝑧𝑅)
52, 4sylan2b 285 . . . 4 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
65exlimivv 1876 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
76abssi 3203 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)} ⊆ 𝑅
81, 7eqsstri 3160 1 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wex 1472  wcel 2128  {cab 2143  wss 3102  cop 3563   class class class wbr 3965  {copab 4024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-br 3966  df-opab 4026
This theorem is referenced by: (None)
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