Theorem dfrel4 6166

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6921 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)

Hypotheses

Ref	Expression
dfrel4.1	⊢ Ⅎ𝑥𝑅
dfrel4.2	⊢ Ⅎ𝑦𝑅

Assertion

Ref	Expression
dfrel4	⊢ (Rel 𝑅 ↔ 𝑅 = {⟨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.

Step	Hyp	Ref	Expression
1		dfrel4v 6165	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝑎
3		dfrel4.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
4		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝑏
5	2, 3, 4	nfbr 5156	. . . 4 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
6		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑦𝑎
7		dfrel4.2	. . . . 5 ⊢ Ⅎ𝑦𝑅
8		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑦𝑏
9	6, 7, 8	nfbr 5156	. . . 4 ⊢ Ⅎ𝑦 𝑎𝑅𝑏
10		nfv 1914	. . . 4 ⊢ Ⅎ𝑎 𝑥𝑅𝑦
11		nfv 1914	. . . 4 ⊢ Ⅎ𝑏 𝑥𝑅𝑦
12		breq12 5114	. . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎𝑅𝑏 ↔ 𝑥𝑅𝑦))
13	5, 9, 10, 11, 12	cbvopab 5181	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
14	13	eqeq2i 2743	. 2 ⊢ (𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
15	1, 14	bitri 275	1 ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Syntax hints:   ↔ wb 206   = wceq 1540  Ⅎwnfc 2877   class class class wbr 5109  {copab 5171  Rel wrel 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648

This theorem is referenced by:  feqmptdf  6933