Theorem trreleq 36623

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 36312	. . . 4 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
2		id 22	. . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆)
3	1, 2	sseq12d 3950	. . 3 ⊢ (𝑅 = 𝑆 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (𝑆 ∘ 𝑆) ⊆ 𝑆))
4		releq 5677	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
5	3, 4	anbi12d 630	. 2 ⊢ (𝑅 = 𝑆 → (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ∧ Rel 𝑆)))
6		dftrrel2 36618	. 2 ⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7		dftrrel2 36618	. 2 ⊢ ( TrRel 𝑆 ↔ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ∧ Rel 𝑆))
8	5, 6, 7	3bitr4g 313	1 ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ⊆ wss 3883   ∘ ccom 5584  Rel wrel 5585   TrRel wtrrel 36275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-trrel 36615

This theorem is referenced by:  eqvreleq  36642