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Theorem dfrel4 6042
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6718 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 6041 . 2 (Rel 𝑅𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2 nfcv 2977 . . . . 5 𝑥𝑎
3 dfrel4.1 . . . . 5 𝑥𝑅
4 nfcv 2977 . . . . 5 𝑥𝑏
52, 3, 4nfbr 5105 . . . 4 𝑥 𝑎𝑅𝑏
6 nfcv 2977 . . . . 5 𝑦𝑎
7 dfrel4.2 . . . . 5 𝑦𝑅
8 nfcv 2977 . . . . 5 𝑦𝑏
96, 7, 8nfbr 5105 . . . 4 𝑦 𝑎𝑅𝑏
10 nfv 1906 . . . 4 𝑎 𝑥𝑅𝑦
11 nfv 1906 . . . 4 𝑏 𝑥𝑅𝑦
12 breq12 5063 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑅𝑏𝑥𝑅𝑦))
135, 9, 10, 11, 12cbvopab 5129 . . 3 {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
1413eqeq2i 2834 . 2 (𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
151, 14bitri 276 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wnfc 2961   class class class wbr 5058  {copab 5120  Rel wrel 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557
This theorem is referenced by:  feqmptdf  6729
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