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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 6329 | . 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 3834 | . 2 ⊢ ∪ 𝐴 = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
4 | 1, 3 | eqtr4i 2213 | 1 ⊢ recs(𝐹) = ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 {cab 2175 ∀wral 2468 ∃wrex 2469 ∪ cuni 3824 Oncon0 4381 ↾ cres 4646 Fn wfn 5230 ‘cfv 5235 recscrecs 6328 |
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-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-uni 3825 df-recs 6329 |
This theorem is referenced by: tfrlem6 6340 tfrlem7 6341 tfrlem8 6342 tfrlem9 6343 tfrlemibfn 6352 tfrlemiubacc 6354 tfrlemi14d 6357 tfrexlem 6358 |
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