Theorem nfttc 37097

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 37093	. 2 ⊢ TC+ 𝐴 = ∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
2		nfttc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2924	. . 3 ⊢ Ⅎ𝑥∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
4	2, 3	nfiun 4986	. 2 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ∪ (rec((𝑧 ∈ V ↦ ∪ 𝑧), {𝑦}) “ ω)
5	1, 4	nfcxfr 2922	1 ⊢ Ⅎ𝑥TC+ 𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2909  Vcvv 3453  {csn 4587  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5190   “ cima 5662  ωcom 7865  reccrdg 8401  TC+ cttc 37092

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 2215  ax-ext 2734

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 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-iun 4956  df-ttc 37093

This theorem is used by:  csbttc  37115