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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 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) | 
| Colors of variables: wff setvar class | 
| 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 1oc1o 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 | 
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