Theorem nfttrcld 9667

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 9665	. 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 1936	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1936	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfv 1936	. . . 4 ⊢ Ⅎ𝑛𝜑
5		nfcvd 2927	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o))
6		nfv 1936	. . . . 5 ⊢ Ⅎ𝑓𝜑
7		nfvd 1937	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8		nfvd 1937	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧))
9		nfv 1936	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
10		nfcvd 2927	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛)
11		nfcvd 2927	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎))
12		nfttrcld.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅)
13		nfcvd 2927	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎))
14	11, 12, 13	nfbrd 5148	. . . . . . 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 1920	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
17	6, 16	nfexd 2363	. . . 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 5170	. 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 2925	1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801  Ⅎwnfc 2911  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903  ∅c0 4287   class class class wbr 5102  {copab 5164  suc csuc 6350   Fn wfn 6518  ‘cfv 6523  ωcom 7848  1_oc1o 8432  t++cttrcl 9664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-ttrcl 9665

This theorem is referenced by:  nfttrcl  9668