Theorem dfrel6 36560

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 36559	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2		dfres3 5908	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
3	2	eqeq1i 2741	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	1, 3	bitri 275	1 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Syntax hints:   ↔ wb 205   = wceq 1539   ∩ cin 3891   × cxp 5598  dom cdm 5600  ran crn 5601   ↾ cres 5602  Rel wrel 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608  df-dm 5610  df-rn 5611  df-res 5612

This theorem is referenced by:  cnvref4  36563  elrels6  36704  dfrefrel2  36729  dfcnvrefrel2  36744  dfsymrel2  36763  dftrrel2  36791