Theorem ttceq 36853

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 4967	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
2		df-ttc 36852	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
3		df-ttc 36852	. 2 ⊢ TC+ 𝐵 = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
4	1, 2, 3	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)

Syntax hints:   → wi 4   = wceq 1561  Vcvv 3455  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5182   “ cima 5651  ωcom 7846  reccrdg 8380  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3088  df-v 3457  df-ss 3922  df-iun 4952  df-ttc 36852

This theorem is referenced by:  ttceqi  36854  ttceqd  36855  ttc00  36873  ttciunun  36876  ttcsng  36884