Theorem dfrel5 37818

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 6193	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		resdm2 6235	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = ◡◡𝑅
3	2	eqeq1i 2733	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ ◡◡𝑅 = 𝑅)
4	1, 3	bitr4i 278	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Syntax hints:   ↔ wb 205   = wceq 1534  ^◡ccnv 5677  dom cdm 5678   ↾ cres 5680  Rel wrel 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690

This theorem is referenced by:  dfrel6  37819  cnvresrn  37820  elrels5  37961