Definition df-trpred 30962

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 30974 and trpredmintr 30975). 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 ↦ ∪ 𝑦 ∈ 𝑎 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3		cX	. . 3 class 𝑋
4	1, 2, 3	ctrpred 30961	. 2 class TrPred(𝑅, 𝐴, 𝑋)
5		va	. . . . . . 7 setvar 𝑎
6		cvv 3173	. . . . . . 7 class V
7		vy	. . . . . . . 8 setvar 𝑦
8	5	cv 1474	. . . . . . . 8 class 𝑎
9	7	cv 1474	. . . . . . . . 9 class 𝑦
10	1, 2, 9	cpred 5596	. . . . . . . 8 class Pred(𝑅, 𝐴, 𝑦)
11	7, 8, 10	ciun 4455	. . . . . . 7 class ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)
12	5, 6, 11	cmpt 4643	. . . . . 6 class (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦))
13	1, 2, 3	cpred 5596	. . . . . 6 class Pred(𝑅, 𝐴, 𝑋)
14	12, 13	crdg 7392	. . . . 5 class rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
15		com 6957	. . . . 5 class ω
16	14, 15	cres 5040	. . . 4 class (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
17	16	crn 5039	. . 3 class ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
18	17	cuni 4372	. 2 class ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
19	4, 18	wceq 1475	1 wff TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)

This definition is referenced by:  dftrpred2  30963  trpredeq1  30964  trpredeq2  30965  trpredeq3  30966  trpredpred  30972  trpredex  30981