Theorem dfrel6 39037

Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)

Assertion

Ref	Expression
dfrel6	⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Proof of Theorem dfrel6

Step	Hyp	Ref	Expression
1		dfrel5 39036	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2		dfres3 5988	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
3	2	eqeq1i 2771	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	1, 3	bitri 278	1 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∩ cin 3907   × cxp 5664  dom cdm 5666  ran crn 5667   ↾ cres 5668  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678

This theorem is used by:  cnvref4  39040  elrels6  39135  dfrefrel2  39285  dfcnvrefrel2  39300  dfsymrel2  39323  dftrrel2  39351