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Theorem trreleq 37964
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 37625 . . . 4 (𝑅 = 𝑆 → (𝑅𝑅) = (𝑆𝑆))
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 4010 . . 3 (𝑅 = 𝑆 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑆𝑆) ⊆ 𝑆))
4 releq 5769 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 630 . 2 (𝑅 = 𝑆 → (((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆𝑆) ⊆ 𝑆 ∧ Rel 𝑆)))
6 dftrrel2 37959 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7 dftrrel2 37959 . 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 205  wa 395   = wceq 1533  wss 3943  ccom 5673  Rel wrel 5674   TrRel wtrrel 37570
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-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-trrel 37956
This theorem is referenced by:  eqvreleq  37984
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