Theorem nfttrcld 9692

Description: Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)

Hypothesis

Ref	Expression
nfttrcld.1	⊢ (𝜑 → Ⅎ𝑥𝑅)

Assertion

Ref	Expression
nfttrcld	⊢ (𝜑 → Ⅎ𝑥t++𝑅)

Proof of Theorem nfttrcld

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

Step	Hyp	Ref	Expression
1		df-ttrcl 9690	. 2 ⊢ t++𝑅 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
2		nfv 1947	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1947	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfv 1947	. . . 4 ⊢ Ⅎ𝑛𝜑
5		nfcvd 2925	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o))
6		nfv 1947	. . . . 5 ⊢ Ⅎ𝑓𝜑
7		nfvd 1948	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8		nfvd 1948	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧))
9		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
10		nfcvd 2925	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛)
11		nfcvd 2925	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎))
12		nfttrcld.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅)
13		nfcvd 2925	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎))
14	11, 12, 13	nfbrd 5155	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
15	9, 10, 14	nfraldw 3309	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
16	7, 8, 15	nf3and 1931	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
17	6, 16	nfexd 2361	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
18	4, 5, 17	nfrexdw 3310	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
19	2, 3, 18	nfopabd 5177	. 2 ⊢ (𝜑 → Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))})
20	1, 19	nfcxfrd 2923	1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812  Ⅎwnfc 2909  ∀wral 3078  ∃wrex 3088   ∖ cdif 3899  ∅c0 4282   class class class wbr 5107  {copab 5171  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7865  1_oc1o 8451  t++cttrcl 9689

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-3an 1105  df-tru 1573  df-fal 1583  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-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-ttrcl 9690

This theorem is used by:  nfttrcl  9693