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Mirrors > Home > MPE Home > Th. List > nfttrcld | Structured version Visualization version GIF version |
Description: Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.) |
Ref | Expression |
---|---|
nfttrcld.1 | ⊢ (𝜑 → Ⅎ𝑥𝑅) |
Ref | Expression |
---|---|
nfttrcld | ⊢ (𝜑 → Ⅎ𝑥t++𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ttrcl 9739 | . 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 1909 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1909 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑛𝜑 | |
5 | nfcvd 2900 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o)) | |
6 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑓𝜑 | |
7 | nfvd 1910 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛) | |
8 | nfvd 1910 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧)) | |
9 | nfv 1909 | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
10 | nfcvd 2900 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛) | |
11 | nfcvd 2900 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)) | |
12 | nfttrcld.1 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
13 | nfcvd 2900 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎)) | |
14 | 11, 12, 13 | nfbrd 5198 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) |
15 | 9, 10, 14 | nfraldw 3304 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) |
16 | 7, 8, 15 | nf3and 1893 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) |
17 | 6, 16 | nfexd 2317 | . . . 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 3305 | . . 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 5220 | . 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 2898 | 1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 Ⅎwnfc 2879 ∀wral 3058 ∃wrex 3067 ∖ cdif 3946 ∅c0 4326 class class class wbr 5152 {copab 5214 suc csuc 6376 Fn wfn 6548 ‘cfv 6553 ωcom 7876 1oc1o 8486 t++cttrcl 9738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-ttrcl 9739 |
This theorem is referenced by: nfttrcl 9742 |
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