Theorem dfrel4 6147

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6890 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 6146	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏})
2		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑥𝑎
3		dfrel4.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
4		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑥𝑏
5	2, 3, 4	nfbr 5143	. . . 4 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
6		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑦𝑎
7		dfrel4.2	. . . . 5 ⊢ Ⅎ𝑦𝑅
8		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑦𝑏
9	6, 7, 8	nfbr 5143	. . . 4 ⊢ Ⅎ𝑦 𝑎𝑅𝑏
10		nfv 1915	. . . 4 ⊢ Ⅎ𝑎 𝑥𝑅𝑦
11		nfv 1915	. . . 4 ⊢ Ⅎ𝑏 𝑥𝑅𝑦
12		breq12 5101	. . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎𝑅𝑏 ↔ 𝑥𝑅𝑦))
13	5, 9, 10, 11, 12	cbvopab 5168	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎𝑅𝑏} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
14	13	eqeq2i 2747	. 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 1541  Ⅎwnfc 2881   class class class wbr 5096  {copab 5158  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630

This theorem is referenced by:  feqmptdf  6902