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Definition df-recs 5830
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 5829 . 2 class recs(𝐹)
3 vf . . . . . . . 8 setvar f
43cv 1222 . . . . . . 7 class f
5 vx . . . . . . . 8 setvar x
65cv 1222 . . . . . . 7 class x
74, 6wfn 4812 . . . . . 6 wff f Fn x
8 vy . . . . . . . . . 10 setvar y
98cv 1222 . . . . . . . . 9 class y
109, 4cfv 4817 . . . . . . . 8 class (fy)
114, 9cres 4262 . . . . . . . . 9 class (fy)
1211, 1cfv 4817 . . . . . . . 8 class (𝐹‘(fy))
1310, 12wceq 1223 . . . . . . 7 wff (fy) = (𝐹‘(fy))
1413, 8, 6wral 2275 . . . . . 6 wff y x (fy) = (𝐹‘(fy))
157, 14wa 97 . . . . 5 wff (f Fn x y x (fy) = (𝐹‘(fy)))
16 con0 4038 . . . . 5 class On
1715, 5, 16wrex 2276 . . . 4 wff x On (f Fn x y x (fy) = (𝐹‘(fy)))
1817, 3cab 1999 . . 3 class {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
1918cuni 3543 . 2 class {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
202, 19wceq 1223 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  5831  nfrecs  5832  recsfval  5841  tfrlem9  5845
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