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Theorem dfrel4 5486
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6132 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 5485 . 2 (Rel 𝑅𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2 nfcv 2746 . . . . 5 𝑥𝑎
3 dfrel4.1 . . . . 5 𝑥𝑅
4 nfcv 2746 . . . . 5 𝑥𝑏
52, 3, 4nfbr 4619 . . . 4 𝑥 𝑎𝑅𝑏
6 nfcv 2746 . . . . 5 𝑦𝑎
7 dfrel4.2 . . . . 5 𝑦𝑅
8 nfcv 2746 . . . . 5 𝑦𝑏
96, 7, 8nfbr 4619 . . . 4 𝑦 𝑎𝑅𝑏
10 nfv 1828 . . . 4 𝑎 𝑥𝑅𝑦
11 nfv 1828 . . . 4 𝑏 𝑥𝑅𝑦
12 breq12 4578 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑅𝑏𝑥𝑅𝑦))
135, 9, 10, 11, 12cbvopab 4643 . . 3 {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
1413eqeq2i 2617 . 2 (𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
151, 14bitri 262 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wnfc 2733   class class class wbr 4573  {copab 4632  Rel wrel 5029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030  df-rel 5031  df-cnv 5032
This theorem is referenced by:  feqmptdf  6142
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