Theorem dfrel3 6145

Description: Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Assertion

Ref	Expression
dfrel3	⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Proof of Theorem dfrel3

Step	Hyp	Ref	Expression
1		dfrel2 6136	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		cnvcnv2 6140	. . 3 ⊢ ◡◡𝑅 = (𝑅 ↾ V)
3	2	eqeq1i 2736	. 2 ⊢ (◡◡𝑅 = 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
4	1, 3	bitri 275	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Syntax hints:   ↔ wb 206   = wceq 1541  Vcvv 3436  ^◡ccnv 5615   ↾ cres 5618  Rel wrel 5621

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-res 5628

This theorem is referenced by:  elid  6146  cocnvcnv2  6206  f1ovi  6802  ttrclco  9608