Theorem nfttc 37457

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 37453	. 2 ⊢ TC+ 𝐴 = ∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
2		nfttc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2926	. . 3 ⊢ Ⅎ𝑥∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
4	2, 3	nfiun 4985	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
5	1, 4	nfcxfr 2924	1 ⊢ Ⅎ𝑥TC+ 𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2911  Vcvv 3454  {csn 4588  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189   “ cima 5658  ωcom 7901  reccrdg 8445  TC+ cttc 37452

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-iun 4956  df-ttc 37453

This theorem is used by:  csbttc  37475