Theorem dfrel5 35618

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 6046	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		resdm2 6088	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = ◡◡𝑅
3	2	eqeq1i 2826	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ ◡◡𝑅 = 𝑅)
4	1, 3	bitr4i 280	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)

Syntax hints:   ↔ wb 208   = wceq 1537  ^◡ccnv 5554  dom cdm 5555   ↾ cres 5557  Rel wrel 5560

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567

This theorem is referenced by:  dfrel6  35619  cnvresrn  35620  elrels5  35744