Theorem cvbtrcl 14965

Description: Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)

Assertion

Ref	Expression
cvbtrcl	⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)}

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cvbtrcl

Step	Hyp	Ref	Expression
1		trcleq2lem 14964	. 2 ⊢ (𝑥 = 𝑦 → ((𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)))
2	1	cbvabv 2800	1 ⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)}

Syntax hints:   ∧ wa 395   = wceq 1540  {cab 2708   ⊆ wss 3917   ∘ ccom 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ss 3934  df-br 5111  df-opab 5173  df-co 5650

This theorem is referenced by: (None)