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Theorem dftrrel2 36428
Description: Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
dftrrel2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))

Proof of Theorem dftrrel2
StepHypRef Expression
1 df-trrel 36425 . 2 ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 36219 . . . . . 6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 219 . . . . 5 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43, 3coeq12d 5733 . . . 4 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (𝑅𝑅))
54, 3sseq12d 3934 . . 3 (Rel 𝑅 → (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ (𝑅𝑅) ⊆ 𝑅))
65pm5.32ri 579 . 2 ((((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
71, 6bitri 278 1 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  cin 3865  wss 3866   × cxp 5549  dom cdm 5551  ran crn 5552  ccom 5555  Rel wrel 5556   TrRel wtrrel 36085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-trrel 36425
This theorem is referenced by:  dftrrel3  36429  eltrrelsrel  36432  trreleq  36433  dfeqvrel2  36440
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