Theorem ttceq 36786

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 4956	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
2		df-ttc 36785	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
3		df-ttc 36785	. 2 ⊢ TC+ 𝐵 = ∪ 𝑥 ∈ 𝐵 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
4	1, 2, 3	3eqtr4g 2812	1 ⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)

Syntax hints:   → wi 4   = wceq 1550  Vcvv 3444  {csn 4572  ∪ cuni 4855  ∪ ciun 4939   ↦ cmpt 5171   “ cima 5639  ωcom 7831  reccrdg 8364  TC+ cttc 36784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rex 3077  df-v 3446  df-ss 3912  df-iun 4941  df-ttc 36785

This theorem is referenced by:  ttceqi  36787  ttceqd  36788  ttc00  36806  ttciunun  36809  ttcsng  36817