Theorem dfrel3 6090

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 6081	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		cnvcnv2 6085	. . 3 ⊢ ◡◡𝑅 = (𝑅 ↾ V)
3	2	eqeq1i 2743	. 2 ⊢ (◡◡𝑅 = 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
4	1, 3	bitri 274	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Syntax hints:   ↔ wb 205   = wceq 1539  Vcvv 3422  ^◡ccnv 5579   ↾ cres 5582  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-res 5592

This theorem is referenced by:  elid  6091  cocnvcnv2  6151  f1ovi  6738  ttrclco  33704