Theorem trcleq1 15022

Description: Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)

Assertion

Ref	Expression
trcleq1	⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})

Distinct variable groups:   𝑅,𝑟   𝑆,𝑟

Proof of Theorem trcleq1

Step	Hyp	Ref	Expression
1		cleq1 15016	1 ⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  {cab 2747   ⊆ wss 3913  ∩ cint 4913   ∘ ccom 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ral 3086  df-rex 3096  df-ss 3930  df-int 4914

This theorem is referenced by:  trclfv  15033