Theorem dfrel5 38713

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

Step	Hyp	Ref	Expression
1		dfrel2 6140	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		resdm2 6182	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = ◡◡𝑅
3	2	eqeq1i 2744	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ ◡◡𝑅 = 𝑅)
4	1, 3	bitr4i 279	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Syntax hints:   ↔ wb 207   = wceq 1547  ^◡ccnv 5617  dom cdm 5618   ↾ cres 5620  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630

This theorem is referenced by:  dfrel6  38714  cnvresrn  38715  elrels5  38811  dfpre4  38847