Theorem nfttc 36661

Description: Bound-variable hypothesis builder for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)

Hypothesis

Ref	Expression
nfttc.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfttc	⊢ Ⅎ𝑥TC+ 𝐴

Proof of Theorem nfttc

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

Step	Hyp	Ref	Expression
1		df-ttc 36657	. 2 ⊢ TC+ 𝐴 = ∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
2		nfttc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2897	. . 3 ⊢ Ⅎ𝑥∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
4	2, 3	nfiun 4955	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
5	1, 4	nfcxfr 2895	1 ⊢ Ⅎ𝑥TC+ 𝐴

Syntax hints:  Ⅎwnfc 2882  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-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3050  df-rex 3060  df-iun 4925  df-ttc 36657

This theorem is referenced by:  csbttc  36679