Theorem ttceq 37027

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 4972	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
2		df-ttc 37026	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
3		df-ttc 37026	. 2 ⊢ TC+ 𝐵 = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
4	1, 2, 3	3eqtr4g 2822	1 ⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Vcvv 3454  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191   “ cima 5663  ωcom 7860  reccrdg 8394  TC+ cttc 37025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456  df-ss 3921  df-iun 4957  df-ttc 37026

This theorem is used by:  ttceqi  37028  ttceqd  37029  ttc00  37047  ttciunun  37050  ttcsng  37058