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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 6389 | . 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 2319 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2319 | . . . . . . . 8 ⊢ Ⅎ𝑥ω | |
4 | nfv 1528 | . . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚 | |
5 | nffrec.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
6 | nfcv 2319 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚) | |
7 | 5, 6 | nffv 5524 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚)) |
8 | 7 | nfcri 2313 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚)) |
9 | 4, 8 | nfan 1565 | . . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) |
10 | 3, 9 | nfrexya 2518 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) |
11 | nfv 1528 | . . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅ | |
12 | nffrec.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐴 | |
13 | 12 | nfcri 2313 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
14 | 11, 13 | nfan 1565 | . . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴) |
15 | 10, 14 | nfor 1574 | . . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)) |
16 | 15 | nfab 2324 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))} |
17 | 2, 16 | nfmpt 4094 | . . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}) |
18 | 17 | nfrecs 6305 | . . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) |
19 | 18, 3 | nfres 4908 | . 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 2316 | 1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 {cab 2163 Ⅎwnfc 2306 ∃wrex 2456 Vcvv 2737 ∅c0 3422 ↦ cmpt 4063 suc csuc 4364 ωcom 4588 dom cdm 4625 ↾ cres 4627 ‘cfv 5215 recscrecs 6302 freccfrec 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-mpt 4065 df-xp 4631 df-res 4637 df-iota 5177 df-fv 5223 df-recs 6303 df-frec 6389 |
This theorem is referenced by: nfseq 10450 |
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