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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 9652 | . 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 1918 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1918 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑛𝜑 | |
5 | nfcvd 2905 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(ω ∖ 1o)) | |
6 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑓𝜑 | |
7 | nfvd 1919 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛) | |
8 | nfvd 1919 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧)) | |
9 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
10 | nfcvd 2905 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝑛) | |
11 | nfcvd 2905 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)) | |
12 | nfttrcld.1 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
13 | nfcvd 2905 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘suc 𝑎)) | |
14 | 11, 12, 13 | nfbrd 5155 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) |
15 | 9, 10, 14 | nfraldw 3291 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) |
16 | 7, 8, 15 | nf3and 1902 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘𝑛) = 𝑧) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) |
17 | 6, 16 | nfexd 2323 | . . . 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 3292 | . . 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 5177 | . 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 2903 | 1 ⊢ (𝜑 → Ⅎ𝑥t++𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 Ⅎwnfc 2884 ∀wral 3061 ∃wrex 3070 ∖ cdif 3911 ∅c0 4286 class class class wbr 5109 {copab 5171 suc csuc 6323 Fn wfn 6495 ‘cfv 6500 ωcom 7806 1oc1o 8409 t++cttrcl 9651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-ttrcl 9652 |
This theorem is referenced by: nfttrcl 9655 |
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