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Mirrors > Home > ILE Home > Th. List > recsfval | GIF version |
Description: Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed 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 |
---|---|
recsfval | ⊢ recs(𝐹) = ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-recs 5943 | . 2 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
2 | tfrlem.1 | . . 3 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
3 | 2 | unieqi 3611 | . 2 ⊢ ∪ 𝐴 = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
4 | 1, 3 | eqtr4i 2104 | 1 ⊢ recs(𝐹) = ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 {cab 2067 ∀wral 2348 ∃wrex 2349 ∪ cuni 3601 Oncon0 4118 ↾ cres 4365 Fn wfn 4917 ‘cfv 4922 recscrecs 5942 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-uni 3602 df-recs 5943 |
This theorem is referenced by: tfrlem6 5955 tfrlem7 5956 tfrlem8 5957 tfrlem9 5958 tfrlemibfn 5965 tfrlemiubacc 5967 tfrlemi14d 5970 tfrexlem 5971 |
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