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 1570  {cab 2741   ⊆ wss 3905  ∩ cint 4912   ∘ ccom 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-ral 3080  df-rex 3090  df-ss 3922  df-int 4913

This theorem is referenced by:  trclfv  15033