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Definition df-recs 6026
Description: Define a function recs(𝐹) on On, the class of ordinal numbers, by transfinite recursion given a rule 𝐹 which sets the next value given all values so far. See df-irdg 6091 for more details on why this definition is desirable. Unlike df-irdg 6091 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See tfri1d 6056 and tfri2d 6057 for the primary contract of this definition.

(Contributed by Stefan O'Rear, 18-Jan-2015.)

Assertion
Ref Expression
df-recs recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Distinct variable group:   𝑓,𝐹,𝑥,𝑦

Detailed syntax breakdown of Definition df-recs
StepHypRef Expression
1 cF . . 3 class 𝐹
21crecs 6025 . 2 class recs(𝐹)
3 vf . . . . . . . 8 setvar 𝑓
43cv 1286 . . . . . . 7 class 𝑓
5 vx . . . . . . . 8 setvar 𝑥
65cv 1286 . . . . . . 7 class 𝑥
74, 6wfn 4978 . . . . . 6 wff 𝑓 Fn 𝑥
8 vy . . . . . . . . . 10 setvar 𝑦
98cv 1286 . . . . . . . . 9 class 𝑦
109, 4cfv 4983 . . . . . . . 8 class (𝑓𝑦)
114, 9cres 4415 . . . . . . . . 9 class (𝑓𝑦)
1211, 1cfv 4983 . . . . . . . 8 class (𝐹‘(𝑓𝑦))
1310, 12wceq 1287 . . . . . . 7 wff (𝑓𝑦) = (𝐹‘(𝑓𝑦))
1413, 8, 6wral 2355 . . . . . 6 wff 𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))
157, 14wa 102 . . . . 5 wff (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
16 con0 4166 . . . . 5 class On
1715, 5, 16wrex 2356 . . . 4 wff 𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
1817, 3cab 2071 . . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1918cuni 3638 . 2 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
202, 19wceq 1287 1 wff recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff set class
This definition is referenced by:  recseq  6027  nfrecs  6028  recsfval  6036  tfrlem9  6040  tfr0dm  6043  tfr1onlemssrecs  6060  tfrcllemssrecs  6073
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