Theorem trpredrec 9415

Description: A transitive predecessor of 𝑋 is either an immediate predecessor of 𝑋 or an immediate predecessor of a transitive predecessor of 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 9404	. 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 7716	. . . 4 ⊢ (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
3		fveq2 6756	. . . . . . . . . . 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 2824	. . . . . . . . . 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 628	. . . . . . . . 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 228	. . . . . . . 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 6217	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
8		fr0g 8237	. . . . . . . . . . 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 2824	. . . . . . . . 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 476	. . . . . . . 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 6756	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 628	. . . . . . . . . . 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 228	. . . . . . . . . 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 6769	. . . . . . . . . . . . . . . . 17 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∈ V
18		trpredlem1 9405	. . . . . . . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . . . . . 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 6217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
22	21	expcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Se 𝐴 → (𝑧 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
23	22	adantl 481	. . . . . . . . . . . . . . . . . . 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 3106	. . . . . . . . . . . . . . . . 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 7779	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑋)
29		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎𝑗
30		nfmpt1 5178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑎(𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦))
31	30, 28	nfrdg 8216	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
32		nfcv 2906	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎ω
33	31, 32	nfres 5882	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
34	33, 29	nffv 6766	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
35		nfcv 2906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑧)
36	34, 35	nfiun 4951	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎∪ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)
37		eqid 2738	. . . . . . . . . . . . . . . . 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 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
39	38	cbviunv 4966	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧)
40		iuneq1 4937	. . . . . . . . . . . . . . . . . 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	eqtrid 2790	. . . . . . . . . . . . . . . . 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 8239	. . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . 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 228	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 4925	. . . . . . . . . . . . 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 4973	. . . . . . . . . . . . . . . . . 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 9397	. . . . . . . . . . . . . . . . . 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	sseqtrrdi 3968	. . . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
53	52	elpredim 6207	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧)
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ω → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧))
55	51, 54	anim12d 608	. . . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . . 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 216	. . . . . . . . . . . 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 3211	. . . . . . 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 961	. . . . . 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 412	. . . . 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 3211	. 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 241	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   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153   Se wse 5533   ↾ cres 5582  Predcpred 6190  suc csuc 6253  ‘cfv 6418  ωcom 7687  reccrdg 8211  TrPredctrpred 9395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-trpred 9396

This theorem is referenced by: (None)