Theorem nfttc 37201

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 37197	. 2 ⊢ TC+ 𝐴 = ∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
2		nfttc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2922	. . 3 ⊢ Ⅎ𝑥∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
4	2, 3	nfiun 4981	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
5	1, 4	nfcxfr 2920	1 ⊢ Ⅎ𝑥TC+ 𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2907  Vcvv 3450  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185   “ cima 5650  ωcom 7860  reccrdg 8395  TC+ cttc 37196

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-iun 4952  df-ttc 37197

This theorem is used by:  csbttc  37219