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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 6156 | . 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} | |
2 | nfcv 2255 | . . . . 5 ⊢ Ⅎ𝑥On | |
3 | nfv 1491 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏 | |
4 | nfcv 2255 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
5 | nfrecs.f | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
6 | nfcv 2255 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐) | |
7 | 5, 6 | nffv 5385 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐)) |
8 | 7 | nfeq2 2267 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) |
9 | 4, 8 | nfralxy 2445 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) |
10 | 3, 9 | nfan 1527 | . . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) |
11 | 2, 10 | nfrexxy 2446 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) |
12 | 11 | nfab 2260 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} |
13 | 12 | nfuni 3708 | . 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} |
14 | 1, 13 | nfcxfr 2252 | 1 ⊢ Ⅎ𝑥recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1314 {cab 2101 Ⅎwnfc 2242 ∀wral 2390 ∃wrex 2391 ∪ cuni 3702 Oncon0 4245 ↾ cres 4501 Fn wfn 5076 ‘cfv 5081 recscrecs 6155 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-iota 5046 df-fv 5089 df-recs 6156 |
This theorem is referenced by: nffrec 6247 |
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