Definition df-trpred 9396

Description: Define the transitive predecessors of a class 𝑋 under a relation 𝑅 in a class 𝐴. This class can be thought of as the "smallest" class containing all elements of 𝐴 that are linked to 𝑋 by an 𝑅-chain (see trpredtr 9408 and trpredmintr 9409). Definition based on Theorem 8.4 of Don Monk's notes for Advanced Set Theory, which can be found at https://euclid.colorado.edu/~monkd/setth.pdf 9409. (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 9395	. 2 class TrPred(𝑅, 𝐴, 𝑋)
5		va	. . . . . . 7 setvar 𝑎
6		cvv 3422	. . . . . . 7 class V
7		vy	. . . . . . . 8 setvar 𝑦
8	5	cv 1538	. . . . . . . 8 class 𝑎
9	7	cv 1538	. . . . . . . . 9 class 𝑦
10	1, 2, 9	cpred 6190	. . . . . . . 8 class Pred(𝑅, 𝐴, 𝑦)
11	7, 8, 10	ciun 4921	. . . . . . 7 class ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)
12	5, 6, 11	cmpt 5153	. . . . . 6 class (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦))
13	1, 2, 3	cpred 6190	. . . . . 6 class Pred(𝑅, 𝐴, 𝑋)
14	12, 13	crdg 8211	. . . . 5 class rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
15		com 7687	. . . . 5 class ω
16	14, 15	cres 5582	. . . 4 class (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
17	16	crn 5581	. . 3 class ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
18	17	cuni 4836	. 2 class ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
19	4, 18	wceq 1539	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  9397  trpredeq1  9398  trpredeq2  9399  trpredeq3  9400  trpredpred  9406  trpredex  9416