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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 5918 | . 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 2175 | . . . . 5 ⊢ ℲxV | |
3 | nfcv 2175 | . . . . . . . 8 ⊢ Ⅎx𝜔 | |
4 | nfv 1418 | . . . . . . . . 9 ⊢ Ⅎxdom g = suc 𝑚 | |
5 | nffrec.1 | . . . . . . . . . . 11 ⊢ Ⅎx𝐹 | |
6 | nfcv 2175 | . . . . . . . . . . 11 ⊢ Ⅎx(g‘𝑚) | |
7 | 5, 6 | nffv 5128 | . . . . . . . . . 10 ⊢ Ⅎx(𝐹‘(g‘𝑚)) |
8 | 7 | nfcri 2169 | . . . . . . . . 9 ⊢ Ⅎx y ∈ (𝐹‘(g‘𝑚)) |
9 | 4, 8 | nfan 1454 | . . . . . . . 8 ⊢ Ⅎx(dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) |
10 | 3, 9 | nfrexya 2357 | . . . . . . 7 ⊢ Ⅎx∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) |
11 | nfv 1418 | . . . . . . . 8 ⊢ Ⅎxdom g = ∅ | |
12 | nffrec.2 | . . . . . . . . 9 ⊢ ℲxA | |
13 | 12 | nfcri 2169 | . . . . . . . 8 ⊢ Ⅎx y ∈ A |
14 | 11, 13 | nfan 1454 | . . . . . . 7 ⊢ Ⅎx(dom g = ∅ ∧ y ∈ A) |
15 | 10, 14 | nfor 1463 | . . . . . 6 ⊢ Ⅎx(∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A)) |
16 | 15 | nfab 2179 | . . . . 5 ⊢ Ⅎx{y ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))} |
17 | 2, 16 | nfmpt 3840 | . . . 4 ⊢ Ⅎx(g ∈ V ↦ {y ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))}) |
18 | 17 | nfrecs 5863 | . . 3 ⊢ Ⅎxrecs((g ∈ V ↦ {y ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))})) |
19 | 18, 3 | nfres 4557 | . 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 2172 | 1 ⊢ Ⅎxfrec(𝐹, A) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∨ wo 628 = wceq 1242 ∈ wcel 1390 {cab 2023 Ⅎwnfc 2162 ∃wrex 2301 Vcvv 2551 ∅c0 3218 ↦ cmpt 3809 suc csuc 4068 𝜔com 4256 dom cdm 4288 ↾ cres 4290 ‘cfv 4845 recscrecs 5860 freccfrec 5917 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-xp 4294 df-res 4300 df-iota 4810 df-fv 4853 df-recs 5861 df-frec 5918 |
This theorem is referenced by: nfiseq 8898 |
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