Theorem nfttrcld 9750

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 9748	. 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 1914	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1914	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfv 1914	. . . 4 ⊢ Ⅎ𝑛𝜑
5		nfcvd 2906	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o))
6		nfv 1914	. . . . 5 ⊢ Ⅎ𝑓𝜑
7		nfvd 1915	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8		nfvd 1915	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧))
9		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
10		nfcvd 2906	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛)
11		nfcvd 2906	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎))
12		nfttrcld.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅)
13		nfcvd 2906	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎))
14	11, 12, 13	nfbrd 5189	. . . . . . 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 1898	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
17	6, 16	nfexd 2329	. . . 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 5211	. 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 2904	1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779  Ⅎwnfc 2890  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948  ∅c0 4333   class class class wbr 5143  {copab 5205  suc csuc 6386   Fn wfn 6556  ‘cfv 6561  ωcom 7887  1_oc1o 8499  t++cttrcl 9747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-ttrcl 9748

This theorem is referenced by:  nfttrcl  9751