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Theorem trreleq 35936
 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 35625 . . . 4 (𝑅 = 𝑆 → (𝑅𝑅) = (𝑆𝑆))
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 3975 . . 3 (𝑅 = 𝑆 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑆𝑆) ⊆ 𝑆))
4 releq 5628 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 633 . 2 (𝑅 = 𝑆 → (((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆)))
6 dftrrel2 35931 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7 dftrrel2 35931 . 2 ( TrRel 𝑆 ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆))
85, 6, 73bitr4g 317 1 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ⊆ wss 3908   ∘ ccom 5536  Rel wrel 5537   TrRel wtrrel 35586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-trrel 35928 This theorem is referenced by:  eqvreleq  35955
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