Theorem dfrel6 36461

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 36460	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2		dfres3 5893	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
3	2	eqeq1i 2744	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	1, 3	bitri 274	1 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Syntax hints:   ↔ wb 205   = wceq 1541   ∩ cin 3890   × cxp 5586  dom cdm 5588  ran crn 5589   ↾ cres 5590  Rel wrel 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-rel 5595  df-cnv 5596  df-dm 5598  df-rn 5599  df-res 5600

This theorem is referenced by:  elrels6  36587  dfrefrel2  36612  dfcnvrefrel2  36625  dfsymrel2  36642  dftrrel2  36670