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Theorem dfrel6 34457
 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
StepHypRef Expression
1 dfrel5 34456 . 2 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2 dfres3 5557 . . 3 (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
32eqeq1i 2766 . 2 ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
41, 3bitri 264 1 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1632   ∩ cin 3715   × cxp 5265  dom cdm 5267  ran crn 5268   ↾ cres 5269  Rel wrel 5272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279 This theorem is referenced by:  elrels6  34582  dfrefrel2  34607  dfcnvrefrel2  34620  dfsymrel2  34637
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