Theorem dfrel5 38374

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 6131	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		resdm2 6173	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = ◡◡𝑅
3	2	eqeq1i 2736	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ ◡◡𝑅 = 𝑅)
4	1, 3	bitr4i 278	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Syntax hints:   ↔ wb 206   = wceq 1541  ^◡ccnv 5610  dom cdm 5611   ↾ cres 5613  Rel wrel 5616

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 5229  ax-nul 5239  ax-pr 5365

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-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623

This theorem is referenced by:  dfrel6  38375  cnvresrn  38376  elrels5  38526