Theorem opabss 4000

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 3998	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)}
2		df-br 3938	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3		eleq1 2203	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
4	3	biimpar 295	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝑧 ∈ 𝑅)
5	2, 4	sylan2b 285	. . . 4 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅)
6	5	exlimivv 1869	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅)
7	6	abssi 3177	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)} ⊆ 𝑅
8	1, 7	eqsstri 3134	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅

Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126   ⊆ wss 3076  ⟨cop 3535   class class class wbr 3937  {copab 3996

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089  df-br 3938  df-opab 3998

This theorem is referenced by: (None)