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Mirrors > Home > ILE Home > Th. List > tfrlem6 | GIF version |
Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfrlem6 | ⊢ Rel recs(𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reluni 4662 | . . 3 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑔 ∈ 𝐴 Rel 𝑔) | |
2 | tfrlem.1 | . . . . 5 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
3 | 2 | tfrlem4 6210 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔) |
4 | funrel 5140 | . . . 4 ⊢ (Fun 𝑔 → Rel 𝑔) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔) |
6 | 1, 5 | mprgbir 2490 | . 2 ⊢ Rel ∪ 𝐴 |
7 | 2 | recsfval 6212 | . . 3 ⊢ recs(𝐹) = ∪ 𝐴 |
8 | 7 | releqi 4622 | . 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴) |
9 | 6, 8 | mpbir 145 | 1 ⊢ Rel recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cab 2125 ∀wral 2416 ∃wrex 2417 ∪ cuni 3736 Oncon0 4285 ↾ cres 4541 Rel wrel 4544 Fun wfun 5117 Fn wfn 5118 ‘cfv 5123 recscrecs 6201 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-recs 6202 |
This theorem is referenced by: tfrlem7 6214 |
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