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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 5920 | . 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 3590 | . 2 ⊢ ∪ 𝐴 = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
4 | 1, 3 | eqtr4i 2063 | 1 ⊢ recs(𝐹) = ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 {cab 2026 ∀wral 2306 ∃wrex 2307 ∪ cuni 3580 Oncon0 4100 ↾ cres 4347 Fn wfn 4897 ‘cfv 4902 recscrecs 5919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-uni 3581 df-recs 5920 |
This theorem is referenced by: tfrlem6 5932 tfrlem7 5933 tfrlem8 5934 tfrlem9 5935 tfrlemibfn 5942 tfrlemiubacc 5944 tfrlemi14d 5947 tfrexlem 5948 |
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