Theorem dfrel3 6186

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 6177	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		cnvcnv2 6181	. . 3 ⊢ ◡◡𝑅 = (𝑅 ↾ V)
3	2	eqeq1i 2736	. 2 ⊢ (◡◡𝑅 = 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
4	1, 3	bitri 274	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Syntax hints:   ↔ wb 205   = wceq 1541  Vcvv 3473  ^◡ccnv 5668   ↾ cres 5671  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-res 5681

This theorem is referenced by:  elid  6187  cocnvcnv2  6246  f1ovi  6859  ttrclco  9695