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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 9746 | . 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 1912 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1912 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑛𝜑 | |
5 | nfcvd 2904 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o)) | |
6 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑓𝜑 | |
7 | nfvd 1913 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛) | |
8 | nfvd 1913 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧)) | |
9 | nfv 1912 | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
10 | nfcvd 2904 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛) | |
11 | nfcvd 2904 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)) | |
12 | nfttrcld.1 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
13 | nfcvd 2904 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎)) | |
14 | 11, 12, 13 | nfbrd 5194 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) |
15 | 9, 10, 14 | nfraldw 3307 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) |
16 | 7, 8, 15 | nf3and 1896 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) |
17 | 6, 16 | nfexd 2328 | . . . 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 3308 | . . 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 5216 | . 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 2902 | 1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 Ⅎwnfc 2888 ∀wral 3059 ∃wrex 3068 ∖ cdif 3960 ∅c0 4339 class class class wbr 5148 {copab 5210 suc csuc 6388 Fn wfn 6558 ‘cfv 6563 ωcom 7887 1oc1o 8498 t++cttrcl 9745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-ttrcl 9746 |
This theorem is referenced by: nfttrcl 9749 |
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