Theorem ttceq 37094

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 4971	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
2		df-ttc 37093	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
3		df-ttc 37093	. 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 1570  Vcvv 3453  {csn 4587  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5190   “ cima 5662  ωcom 7865  reccrdg 8401  TC+ cttc 37092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3455  df-ss 3919  df-iun 4956  df-ttc 37093

This theorem is used by:  ttceqi  37095  ttceqd  37096  ttc00  37114  ttciunun  37117  ttcsng  37125