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Theorem dfrel4 5504
 Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6151 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 5503 . 2 (Rel 𝑅𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2 nfcv 2751 . . . . 5 𝑥𝑎
3 dfrel4.1 . . . . 5 𝑥𝑅
4 nfcv 2751 . . . . 5 𝑥𝑏
52, 3, 4nfbr 4629 . . . 4 𝑥 𝑎𝑅𝑏
6 nfcv 2751 . . . . 5 𝑦𝑎
7 dfrel4.2 . . . . 5 𝑦𝑅
8 nfcv 2751 . . . . 5 𝑦𝑏
96, 7, 8nfbr 4629 . . . 4 𝑦 𝑎𝑅𝑏
10 nfv 1830 . . . 4 𝑎 𝑥𝑅𝑦
11 nfv 1830 . . . 4 𝑏 𝑥𝑅𝑦
12 breq12 4588 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑅𝑏𝑥𝑅𝑦))
135, 9, 10, 11, 12cbvopab 4653 . . 3 {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
1413eqeq2i 2622 . 2 (𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
151, 14bitri 263 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  Ⅎwnfc 2738   class class class wbr 4583  {copab 4642  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  feqmptdf  6161
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