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Theorem opabss 4908
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 4907 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)}
2 df-br 4845 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3 eleq1 2873 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
43biimpar 465 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝑧𝑅)
52, 4sylan2b 583 . . . 4 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
65exlimivv 2023 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧𝑅)
76abssi 3874 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)} ⊆ 𝑅
81, 7eqsstri 3832 1 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wss 3769  cop 4376   class class class wbr 4844  {copab 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-in 3776  df-ss 3783  df-br 4845  df-opab 4907
This theorem is referenced by:  aceq3lem  9226  fullfunc  16770  fthfunc  16771  isfull  16774  isfth  16778
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