Theorem trpredrec 31738

Description: If 𝑌 is an 𝑅, 𝐴 transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between 𝑌 and 𝑋. (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
trpredrec	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))

Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋   𝑧,𝑌

Proof of Theorem trpredrec

Dummy variables 𝑎 𝑖 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltrpred 31726	. 2 ⊢ (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) ↔ ∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖))
2		nn0suc 7090	. . . 4 ⊢ (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
3		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))
4	3	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝑖 = ∅ → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ↔ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅)))
5	4	anbi2d 740	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))))
6	5	biimpd 219	. . . . . . . 8 ⊢ (𝑖 = ∅ → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))))
7		setlikespec 5701	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
8		fr0g 7531	. . . . . . . . . . 11 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
9	7, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
10	9	eleq2d 2687	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
11	10	biimpa 501	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅)) → 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))
12	6, 11	syl6com 37	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (𝑖 = ∅ → 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
13		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))
14	13	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑖 = suc 𝑗 → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ↔ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)))
15	14	anbi2d 740	. . . . . . . . . . 11 ⊢ (𝑖 = suc 𝑗 → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))))
16	15	biimpd 219	. . . . . . . . . 10 ⊢ (𝑖 = suc 𝑗 → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))))
17		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∈ V
18		trpredlem1 31727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐴)
19	7, 18	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐴)
20	19	sseld 3602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑧 ∈ 𝐴))
21		setlikespec 5701	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
22	21	expcom 451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Se 𝐴 → (𝑧 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
23	22	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
24	20, 23	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → Pred(𝑅, 𝐴, 𝑧) ∈ V))
25	24	ralrimiv 2965	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V)
26		iunexg 7143	. . . . . . . . . . . . . . . . 17 ⊢ ((((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∈ V ∧ ∀𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V) → ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V)
27	17, 25, 26	sylancr 695	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V)
28		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑋)
29		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎𝑗
30		nfmpt1 4747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑎(𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦))
31	30, 28	nfrdg 7510	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
32		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎ω
33	31, 32	nfres 5398	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
34	33, 29	nffv 6198	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
35		nfcv 2764	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑧)
36	34, 35	nfiun 4548	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)
37		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
38		predeq3 5684	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
39	38	cbviunv 4559	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧)
40		iuneq1 4534	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → ∪ 𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) = ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
41	39, 40	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
42	28, 29, 36, 37, 41	frsucmpt 7533	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ω ∧ ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
43	27, 42	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ω ∧ (𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
44	43	eleq2d 2687	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ω ∧ (𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴)) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ↔ 𝑌 ∈ ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)))
45	44	biimpd 219	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ω ∧ (𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴)) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) → 𝑌 ∈ ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)))
46	45	expimpd 629	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ω → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)) → 𝑌 ∈ ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)))
47		eliun 4524	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ↔ ∃𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)𝑌 ∈ Pred(𝑅, 𝐴, 𝑧))
48		ssiun2 4563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ ∪ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
49		dftrpred2 31719	. . . . . . . . . . . . . . . . . 18 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
50	48, 49	syl6sseqr 3652	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ TrPred(𝑅, 𝐴, 𝑋))
51	50	sseld 3602	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ω → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑧 ∈ TrPred(𝑅, 𝐴, 𝑋)))
52		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
53	52	elpredim 5692	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧)
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ω → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧))
55	51, 54	anim12d 586	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ω → ((𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑧)) → (𝑧 ∈ TrPred(𝑅, 𝐴, 𝑋) ∧ 𝑌𝑅𝑧)))
56	55	reximdv2 3014	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ω → (∃𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
57	56	com12 32	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
58	47, 57	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ ∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
59	46, 58	syl6com 37	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)) → (𝑗 ∈ ω → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
60	59	pm2.43d 53	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)) → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
61	16, 60	syl6com 37	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (𝑖 = suc 𝑗 → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
62	61	com23 86	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (𝑗 ∈ ω → (𝑖 = suc 𝑗 → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
63	62	rexlimdv 3030	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
64	12, 63	orim12d 883	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
65	64	ex 450	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))))
66	65	com23 86	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))))
67	2, 66	syl5 34	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑖 ∈ ω → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))))
68	67	rexlimdv 3030	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
69	1, 68	syl5bi 232	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   Se wse 5071   ↾ cres 5116  Predcpred 5679  suc csuc 5725  ‘cfv 5888  ωcom 7065  reccrdg 7505  TrPredctrpred 31717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-trpred 31718

This theorem is referenced by: (None)