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Theorem dfrel4 6088
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6821 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
Hypotheses
Ref Expression
dfrel4.1 𝑥𝑅
dfrel4.2 𝑦𝑅
Assertion
Ref Expression
dfrel4 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem dfrel4
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrel4v 6087 . 2 (Rel 𝑅𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2 nfcv 2907 . . . . 5 𝑥𝑎
3 dfrel4.1 . . . . 5 𝑥𝑅
4 nfcv 2907 . . . . 5 𝑥𝑏
52, 3, 4nfbr 5121 . . . 4 𝑥 𝑎𝑅𝑏
6 nfcv 2907 . . . . 5 𝑦𝑎
7 dfrel4.2 . . . . 5 𝑦𝑅
8 nfcv 2907 . . . . 5 𝑦𝑏
96, 7, 8nfbr 5121 . . . 4 𝑦 𝑎𝑅𝑏
10 nfv 1917 . . . 4 𝑎 𝑥𝑅𝑦
11 nfv 1917 . . . 4 𝑏 𝑥𝑅𝑦
12 breq12 5079 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑅𝑏𝑥𝑅𝑦))
135, 9, 10, 11, 12cbvopab 5146 . . 3 {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
1413eqeq2i 2751 . 2 (𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
151, 14bitri 274 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  wnfc 2887   class class class wbr 5074  {copab 5136  Rel wrel 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-xp 5591  df-rel 5592  df-cnv 5593
This theorem is referenced by:  feqmptdf  6832
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