Theorem nfttrcld 9681

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 9679	. 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 1941	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1941	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfv 1941	. . . 4 ⊢ Ⅎ𝑛𝜑
5		nfcvd 2932	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o))
6		nfv 1941	. . . . 5 ⊢ Ⅎ𝑓𝜑
7		nfvd 1942	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8		nfvd 1942	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧))
9		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
10		nfcvd 2932	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛)
11		nfcvd 2932	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎))
12		nfttrcld.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅)
13		nfcvd 2932	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎))
14	11, 12, 13	nfbrd 5161	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
15	9, 10, 14	nfraldw 3316	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
16	7, 8, 15	nf3and 1925	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
17	6, 16	nfexd 2368	. . . 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 3317	. . 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 5183	. 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 2930	1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806  Ⅎwnfc 2916  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910  ∅c0 4294   class class class wbr 5113  {copab 5177  suc csuc 6365   Fn wfn 6534  ‘cfv 6539  ωcom 7864  1_oc1o 8448  t++cttrcl 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-ttrcl 9679

This theorem is referenced by:  nfttrcl  9682