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Definition df-recs 6170
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 6235 for more details on why this definition is desirable. Unlike df-irdg 6235 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 6200 and tfri2d 6201 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 6169 . 2 class recs(𝐹)
3 vf . . . . . . . 8 setvar 𝑓
43cv 1315 . . . . . . 7 class 𝑓
5 vx . . . . . . . 8 setvar 𝑥
65cv 1315 . . . . . . 7 class 𝑥
74, 6wfn 5088 . . . . . 6 wff 𝑓 Fn 𝑥
8 vy . . . . . . . . . 10 setvar 𝑦
98cv 1315 . . . . . . . . 9 class 𝑦
109, 4cfv 5093 . . . . . . . 8 class (𝑓𝑦)
114, 9cres 4511 . . . . . . . . 9 class (𝑓𝑦)
1211, 1cfv 5093 . . . . . . . 8 class (𝐹‘(𝑓𝑦))
1310, 12wceq 1316 . . . . . . 7 wff (𝑓𝑦) = (𝐹‘(𝑓𝑦))
1413, 8, 6wral 2393 . . . . . 6 wff 𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))
157, 14wa 103 . . . . 5 wff (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
16 con0 4255 . . . . 5 class On
1715, 5, 16wrex 2394 . . . 4 wff 𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
1817, 3cab 2103 . . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1918cuni 3706 . 2 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
202, 19wceq 1316 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  6171  nfrecs  6172  recsfval  6180  tfrlem9  6184  tfr0dm  6187  tfr1onlemssrecs  6204  tfrcllemssrecs  6217
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