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Theorem dftrrel2 38081
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 38078 . 2 ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 37851 . . . . . 6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 215 . . . . 5 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43, 3coeq12d 5871 . . . 4 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (𝑅𝑅))
54, 3sseq12d 4015 . . 3 (Rel 𝑅 → (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ (𝑅𝑅) ⊆ 𝑅))
65pm5.32ri 574 . 2 ((((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
71, 6bitri 274 1 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  cin 3948  wss 3949   × cxp 5680  dom cdm 5682  ran crn 5683  ccom 5686  Rel wrel 5687   TrRel wtrrel 37696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-trrel 38078
This theorem is referenced by:  dftrrel3  38082  eltrrelsrel  38085  trreleq  38086  dfeqvrel2  38094
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