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Theorem trreleq 34875
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 34563 . . . 4 (𝑅 = 𝑆 → (𝑅𝑅) = (𝑆𝑆))
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 3858 . . 3 (𝑅 = 𝑆 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑆𝑆) ⊆ 𝑆))
4 releq 5435 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 626 . 2 (𝑅 = 𝑆 → (((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆)))
6 dftrrel2 34870 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7 dftrrel2 34870 . 2 ( TrRel 𝑆 ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆))
85, 6, 73bitr4g 306 1 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wss 3797  ccom 5345  Rel wrel 5346   TrRel wtrrel 34538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-trrel 34867
This theorem is referenced by:  eqvreleq  34891
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