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Definition df-trpred 31842
Description: Define the transitive predecessors of a class 𝑋 under a relationship 𝑅 and a class 𝐴. This class can be thought of as the "smallest" class containing all elements of 𝐴 that are linked to 𝑋 by a chain of 𝑅 relationships (see trpredtr 31854 and trpredmintr 31855). Definition based off of Lemma 4.2 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory (check The Internet Archive for it now as Prof. Monk appears to have rewritten his website). (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
df-trpred TrPred(𝑅, 𝐴, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
Distinct variable groups:   𝑅,𝑎,𝑦   𝐴,𝑎,𝑦   𝑋,𝑎,𝑦

Detailed syntax breakdown of Definition df-trpred
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
3 cX . . 3 class 𝑋
41, 2, 3ctrpred 31841 . 2 class TrPred(𝑅, 𝐴, 𝑋)
5 va . . . . . . 7 setvar 𝑎
6 cvv 3231 . . . . . . 7 class V
7 vy . . . . . . . 8 setvar 𝑦
85cv 1522 . . . . . . . 8 class 𝑎
97cv 1522 . . . . . . . . 9 class 𝑦
101, 2, 9cpred 5717 . . . . . . . 8 class Pred(𝑅, 𝐴, 𝑦)
117, 8, 10ciun 4552 . . . . . . 7 class 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)
125, 6, 11cmpt 4762 . . . . . 6 class (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦))
131, 2, 3cpred 5717 . . . . . 6 class Pred(𝑅, 𝐴, 𝑋)
1412, 13crdg 7550 . . . . 5 class rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
15 com 7107 . . . . 5 class ω
1614, 15cres 5145 . . . 4 class (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
1716crn 5144 . . 3 class ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
1817cuni 4468 . 2 class ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
194, 18wceq 1523 1 wff TrPred(𝑅, 𝐴, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
Colors of variables: wff setvar class
This definition is referenced by:  dftrpred2  31843  trpredeq1  31844  trpredeq2  31845  trpredeq3  31846  trpredpred  31852  trpredex  31861
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