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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 5978 | . 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 2178 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2178 | . . . . . . . 8 ⊢ Ⅎ𝑥ω | |
4 | nfv 1421 | . . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚 | |
5 | nffrec.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
6 | nfcv 2178 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚) | |
7 | 5, 6 | nffv 5185 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚)) |
8 | 7 | nfcri 2172 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚)) |
9 | 4, 8 | nfan 1457 | . . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) |
10 | 3, 9 | nfrexya 2363 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) |
11 | nfv 1421 | . . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅ | |
12 | nffrec.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐴 | |
13 | 12 | nfcri 2172 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
14 | 11, 13 | nfan 1457 | . . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴) |
15 | 10, 14 | nfor 1466 | . . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)) |
16 | 15 | nfab 2182 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))} |
17 | 2, 16 | nfmpt 3849 | . . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}) |
18 | 17 | nfrecs 5922 | . . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) |
19 | 18, 3 | nfres 4614 | . 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 2175 | 1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∨ wo 629 = wceq 1243 ∈ wcel 1393 {cab 2026 Ⅎwnfc 2165 ∃wrex 2307 Vcvv 2557 ∅c0 3224 ↦ cmpt 3818 suc csuc 4102 ωcom 4313 dom cdm 4345 ↾ cres 4347 ‘cfv 4902 recscrecs 5919 freccfrec 5977 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-xp 4351 df-res 4357 df-iota 4867 df-fv 4910 df-recs 5920 df-frec 5978 |
This theorem is referenced by: nfiseq 9218 |
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