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Definition df-recs 5838
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. If we were assuming the law of the excluded middle, we would then build on top of that a form of recursion which has separate cases for the empty set, successor ordinals, and limit ordinals. This version allows the update rule to use all previous values, which is why it is described as "strong".

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

Assertion
Ref Expression
df-recs recs(𝐹) = {fx On (f Fn x y x (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 5837 . 2 class recs(𝐹)
3 vf . . . . . . . 8 setvar f
43cv 1225 . . . . . . 7 class f
5 vx . . . . . . . 8 setvar x
65cv 1225 . . . . . . 7 class x
74, 6wfn 4820 . . . . . 6 wff f Fn x
8 vy . . . . . . . . . 10 setvar y
98cv 1225 . . . . . . . . 9 class y
109, 4cfv 4825 . . . . . . . 8 class (fy)
114, 9cres 4270 . . . . . . . . 9 class (fy)
1211, 1cfv 4825 . . . . . . . 8 class (𝐹‘(fy))
1310, 12wceq 1226 . . . . . . 7 wff (fy) = (𝐹‘(fy))
1413, 8, 6wral 2280 . . . . . 6 wff y x (fy) = (𝐹‘(fy))
157, 14wa 97 . . . . 5 wff (f Fn x y x (fy) = (𝐹‘(fy)))
16 con0 4045 . . . . 5 class On
1715, 5, 16wrex 2281 . . . 4 wff x On (f Fn x y x (fy) = (𝐹‘(fy)))
1817, 3cab 2004 . . 3 class {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
1918cuni 3550 . 2 class {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
202, 19wceq 1226 1 wff recs(𝐹) = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Colors of variables: wff set class
This definition is referenced by:  recseq  5839  nfrecs  5840  recsfval  5849  tfrlem9  5853
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