Theorem opabss 5136

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.

Step	Hyp	Ref	Expression
1		df-opab 5135	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)}
2		df-br 5073	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3		eleq1 2827	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
4	3	biimpar 478	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝑧 ∈ 𝑅)
5	2, 4	sylan2b 600	. . . 4 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅)
6	5	exlimivv 1939	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅)
7	6	abssi 3999	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)} ⊆ 𝑅
8	1, 7	eqsstri 3961	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅

Syntax hints:   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717   ⊆ wss 3883  ⟨cop 4561   class class class wbr 5072  {copab 5134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ss 3900  df-br 5073  df-opab 5135

This theorem is referenced by:  elopabr  5502  elopabran  5503  mptmpoopabbrd  8022  aceq3lem  10033  fullfunc  17866  fthfunc  17867  isfull  17870  isfth  17874  wksv  29706