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Theorem dfrel4 5554
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6208 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 5553 . 2 (Rel 𝑅𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2 nfcv 2761 . . . . 5 𝑥𝑎
3 dfrel4.1 . . . . 5 𝑥𝑅
4 nfcv 2761 . . . . 5 𝑥𝑏
52, 3, 4nfbr 4669 . . . 4 𝑥 𝑎𝑅𝑏
6 nfcv 2761 . . . . 5 𝑦𝑎
7 dfrel4.2 . . . . 5 𝑦𝑅
8 nfcv 2761 . . . . 5 𝑦𝑏
96, 7, 8nfbr 4669 . . . 4 𝑦 𝑎𝑅𝑏
10 nfv 1840 . . . 4 𝑎 𝑥𝑅𝑦
11 nfv 1840 . . . 4 𝑏 𝑥𝑅𝑦
12 breq12 4628 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑅𝑏𝑥𝑅𝑦))
135, 9, 10, 11, 12cbvopab 4693 . . 3 {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
1413eqeq2i 2633 . 2 (𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
151, 14bitri 264 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wnfc 2748   class class class wbr 4623  {copab 4682  Rel wrel 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092
This theorem is referenced by:  feqmptdf  6218
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