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Theorem dfrel5 35782
 Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
Assertion
Ref Expression
dfrel5 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Proof of Theorem dfrel5
StepHypRef Expression
1 dfrel2 6014 . 2 (Rel 𝑅𝑅 = 𝑅)
2 resdm2 6056 . . 3 (𝑅 ↾ dom 𝑅) = 𝑅
32eqeq1i 2803 . 2 ((𝑅 ↾ dom 𝑅) = 𝑅𝑅 = 𝑅)
41, 3bitr4i 281 1 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ◡ccnv 5519  dom cdm 5520   ↾ cres 5522  Rel wrel 5525 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532 This theorem is referenced by:  dfrel6  35783  cnvresrn  35784  elrels5  35908
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