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Mirrors > Home > ILE Home > Th. List > nffrec | GIF version |
Description: Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
nffrec.1 | ⊢ Ⅎ𝑥𝐹 |
nffrec.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nffrec | ⊢ Ⅎ𝑥frec(𝐹, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frec 6288 | . 2 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) ↾ ω) | |
2 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2281 | . . . . . . . 8 ⊢ Ⅎ𝑥ω | |
4 | nfv 1508 | . . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚 | |
5 | nffrec.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
6 | nfcv 2281 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚) | |
7 | 5, 6 | nffv 5431 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚)) |
8 | 7 | nfcri 2275 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚)) |
9 | 4, 8 | nfan 1544 | . . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) |
10 | 3, 9 | nfrexya 2474 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) |
11 | nfv 1508 | . . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅ | |
12 | nffrec.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐴 | |
13 | 12 | nfcri 2275 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
14 | 11, 13 | nfan 1544 | . . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴) |
15 | 10, 14 | nfor 1553 | . . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)) |
16 | 15 | nfab 2286 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))} |
17 | 2, 16 | nfmpt 4020 | . . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}) |
18 | 17 | nfrecs 6204 | . . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) |
19 | 18, 3 | nfres 4821 | . 2 ⊢ Ⅎ𝑥(recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) ↾ ω) |
20 | 1, 19 | nfcxfr 2278 | 1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∨ wo 697 = wceq 1331 ∈ wcel 1480 {cab 2125 Ⅎwnfc 2268 ∃wrex 2417 Vcvv 2686 ∅c0 3363 ↦ cmpt 3989 suc csuc 4287 ωcom 4504 dom cdm 4539 ↾ cres 4541 ‘cfv 5123 recscrecs 6201 freccfrec 6287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-xp 4545 df-res 4551 df-iota 5088 df-fv 5131 df-recs 6202 df-frec 6288 |
This theorem is referenced by: nfseq 10228 |
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