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Theorem opabss 3812
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 3810 . 2 {⟨x, y⟩ ∣ x𝑅y} = {zxy(z = ⟨x, y x𝑅y)}
2 df-br 3756 . . . . 5 (x𝑅y ↔ ⟨x, y 𝑅)
3 eleq1 2097 . . . . . 6 (z = ⟨x, y⟩ → (z 𝑅 ↔ ⟨x, y 𝑅))
43biimpar 281 . . . . 5 ((z = ⟨x, yx, y 𝑅) → z 𝑅)
52, 4sylan2b 271 . . . 4 ((z = ⟨x, y x𝑅y) → z 𝑅)
65exlimivv 1773 . . 3 (xy(z = ⟨x, y x𝑅y) → z 𝑅)
76abssi 3009 . 2 {zxy(z = ⟨x, y x𝑅y)} ⊆ 𝑅
81, 7eqsstri 2969 1 {⟨x, y⟩ ∣ x𝑅y} ⊆ 𝑅
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wss 2911  cop 3370   class class class wbr 3755  {copab 3808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810
This theorem is referenced by: (None)
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