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Theorem trreleq 38564
Description: Equality theorem for the transitive relation predicate. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
trreleq (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))

Proof of Theorem trreleq
StepHypRef Expression
1 coideq 38228 . . . 4 (𝑅 = 𝑆 → (𝑅𝑅) = (𝑆𝑆))
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 4029 . . 3 (𝑅 = 𝑆 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑆𝑆) ⊆ 𝑆))
4 releq 5789 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 632 . 2 (𝑅 = 𝑆 → (((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆)))
6 dftrrel2 38559 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7 dftrrel2 38559 . 2 ( TrRel 𝑆 ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆))
85, 6, 73bitr4g 314 1 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wss 3963  ccom 5693  Rel wrel 5694   TrRel wtrrel 38177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-trrel 38556
This theorem is referenced by:  eqvreleq  38584
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