Theorem ttceq 36658

Description: Equality theorem for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttceq	⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)

Proof of Theorem ttceq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1 4940	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
2		df-ttc 36657	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
3		df-ttc 36657	. 2 ⊢ TC+ 𝐵 = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
4	1, 2, 3	3eqtr4g 2795	1 ⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)

Syntax hints:   → wi 4   = wceq 1542  Vcvv 3427  {csn 4557  ∪ cuni 4840  ∪ ciun 4923   ↦ cmpt 5155   “ cima 5623  ωcom 7806  reccrdg 8337  TC+ cttc 36656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-v 3429  df-ss 3902  df-iun 4925  df-ttc 36657

This theorem is referenced by:  ttceqi  36659  ttceqd  36660  ttc00  36678  ttciunun  36681  ttcsng  36689