Theorem nfttc 36856

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 36852	. 2 ⊢ TC+ 𝐴 = ∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
2		nfttc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2925	. . 3 ⊢ Ⅎ𝑥∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
4	2, 3	nfiun 4982	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
5	1, 4	nfcxfr 2923	1 ⊢ Ⅎ𝑥TC+ 𝐴

Syntax hints:  Ⅎwnfc 2910  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-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-iun 4952  df-ttc 36852

This theorem is referenced by:  csbttc  36874