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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 6108 | . 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 3685 | . 2 ⊢ ∪ 𝐴 = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
4 | 1, 3 | eqtr4i 2118 | 1 ⊢ recs(𝐹) = ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1296 {cab 2081 ∀wral 2370 ∃wrex 2371 ∪ cuni 3675 Oncon0 4214 ↾ cres 4469 Fn wfn 5044 ‘cfv 5049 recscrecs 6107 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-uni 3676 df-recs 6108 |
This theorem is referenced by: tfrlem6 6119 tfrlem7 6120 tfrlem8 6121 tfrlem9 6122 tfrlemibfn 6131 tfrlemiubacc 6133 tfrlemi14d 6136 tfrexlem 6137 |
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