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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 6281 | . 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} | |
2 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑥On | |
3 | nfv 1521 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏 | |
4 | nfcv 2312 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
5 | nfrecs.f | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
6 | nfcv 2312 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐) | |
7 | 5, 6 | nffv 5504 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐)) |
8 | 7 | nfeq2 2324 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) |
9 | 4, 8 | nfralxy 2508 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) |
10 | 3, 9 | nfan 1558 | . . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) |
11 | 2, 10 | nfrexxy 2509 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) |
12 | 11 | nfab 2317 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} |
13 | 12 | nfuni 3800 | . 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} |
14 | 1, 13 | nfcxfr 2309 | 1 ⊢ Ⅎ𝑥recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 {cab 2156 Ⅎwnfc 2299 ∀wral 2448 ∃wrex 2449 ∪ cuni 3794 Oncon0 4346 ↾ cres 4611 Fn wfn 5191 ‘cfv 5196 recscrecs 6280 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-recs 6281 |
This theorem is referenced by: nffrec 6372 |
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