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Mirrors > Home > ILE Home > Th. List > nfrecs | GIF version |
Description: Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
nfrecs.f | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
nfrecs | ⊢ Ⅎ𝑥recs(𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-recs 6330 | . 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} | |
2 | nfcv 2332 | . . . . 5 ⊢ Ⅎ𝑥On | |
3 | nfv 1539 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏 | |
4 | nfcv 2332 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
5 | nfrecs.f | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
6 | nfcv 2332 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐) | |
7 | 5, 6 | nffv 5544 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐)) |
8 | 7 | nfeq2 2344 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) |
9 | 4, 8 | nfralxy 2528 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) |
10 | 3, 9 | nfan 1576 | . . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) |
11 | 2, 10 | nfrexxy 2529 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) |
12 | 11 | nfab 2337 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} |
13 | 12 | nfuni 3830 | . 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} |
14 | 1, 13 | nfcxfr 2329 | 1 ⊢ Ⅎ𝑥recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 {cab 2175 Ⅎwnfc 2319 ∀wral 2468 ∃wrex 2469 ∪ cuni 3824 Oncon0 4381 ↾ cres 4646 Fn wfn 5230 ‘cfv 5235 recscrecs 6329 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-recs 6330 |
This theorem is referenced by: nffrec 6421 |
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