Theorem trreleq 39265

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

Step	Hyp	Ref	Expression
1		coideq 38847	. . . 4 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
2		id 23	. . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆)
3	1, 2	sseq12d 3978	. . 3 ⊢ (𝑅 = 𝑆 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (𝑆 ∘ 𝑆) ⊆ 𝑆))
4		releq 5767	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
5	3, 4	anbi12d 643	. 2 ⊢ (𝑅 = 𝑆 → (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ∧ Rel 𝑆)))
6		dftrrel2 39260	. 2 ⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7		dftrrel2 39260	. 2 ⊢ ( TrRel 𝑆 ↔ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ∧ Rel 𝑆))
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ⊆ wss 3913   ∘ ccom 5669  Rel wrel 5670   TrRel wtrrel 38797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-trrel 39257

This theorem is referenced by:  eqvreleq  39285