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Theorem dfrel6 35157
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 35156 . 2 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2 dfres3 5746 . . 3 (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
32eqeq1i 2802 . 2 ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
41, 3bitri 276 1 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1525  cin 3864   × cxp 5448  dom cdm 5450  ran crn 5451  cres 5452  Rel wrel 5455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462
This theorem is referenced by:  elrels6  35282  dfrefrel2  35307  dfcnvrefrel2  35320  dfsymrel2  35337  dftrrel2  35365
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