Theorem trpredtr 31730

Description: The transitive predecessors are transitive in 𝑅 and 𝐴 (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
trpredtr	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Proof of Theorem trpredtr

Dummy variables 𝑎 𝑓 𝑖 𝑗 𝑡 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		simplr 792	. . . . . 6 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → 𝑖 ∈ ω)
3		peano2 7086	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
4	2, 3	syl 17	. . . . 5 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → suc 𝑖 ∈ ω)
5		simpr 477	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 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)‘𝑖))
6		ssid 3624	. . . . . . 7 ⊢ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)
7		predeq3 5684	. . . . . . . . . 10 ⊢ (𝑡 = 𝑌 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑌))
8	7	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑡 = 𝑌 → (Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡) ↔ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)))
9	8	rspcev 3309	. . . . . . . 8 ⊢ ((𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)) → ∃𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡))
10		ssiun 4562	. . . . . . . 8 ⊢ (∃𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡) → Pred(𝑅, 𝐴, 𝑌) ⊆ ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)) → Pred(𝑅, 𝐴, 𝑌) ⊆ ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
12	5, 6, 11	sylancl 694	. . . . . 6 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
13		fvex 6201	. . . . . . . 8 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∈ V
14		setlikespec 5701	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
15		trpredlem1 31727	. . . . . . . . . . . . 13 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
16	14, 15	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
17	16	sseld 3602	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → 𝑡 ∈ 𝐴))
18		setlikespec 5701	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑡) ∈ V)
19	18	expcom 451	. . . . . . . . . . . 12 ⊢ (𝑅 Se 𝐴 → (𝑡 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
20	19	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑡 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
21	17, 20	syld 47	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → Pred(𝑅, 𝐴, 𝑡) ∈ V))
22	21	ralrimiv 2965	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
23	22	ad2antrr 762	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ∀𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
24		iunexg 7143	. . . . . . . 8 ⊢ ((((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∈ V ∧ ∀𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V) → ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
25	13, 23, 24	sylancr 695	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
26		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑓Pred(𝑅, 𝐴, 𝑋)
27		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑓𝑖
28		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑓∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡)
29		predeq3 5684	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑡))
30	29	cbviunv 4559	. . . . . . . . . . 11 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑡 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑡)
31		iuneq1 4534	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑓 → ∪ 𝑡 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
32	30, 31	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝑎 = 𝑓 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
33	32	cbvmptv 4750	. . . . . . . . 9 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
34		rdgeq1 7507	. . . . . . . . 9 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡)) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)))
35		reseq1 5390	. . . . . . . . 9 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)) → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
36	33, 34, 35	mp2b 10	. . . . . . . 8 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
37		iuneq1 4534	. . . . . . . 8 ⊢ (𝑓 = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
38	26, 27, 28, 36, 37	frsucmpt 7533	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖) = ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
39	2, 25, 38	syl2anc 693	. . . . . 6 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((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 𝑖) = ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
40	12, 39	sseqtr4d 3642	. . . . 5 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖))
41		fveq2 6191	. . . . . . . . 9 ⊢ (𝑗 = suc 𝑖 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖))
42	41	sseq2d 3633	. . . . . . . 8 ⊢ (𝑗 = suc 𝑖 → (Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ↔ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)))
43	42	rspcev 3309	. . . . . . 7 ⊢ ((suc 𝑖 ∈ ω ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)) → ∃𝑗 ∈ ω Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
44		ssiun 4562	. . . . . . 7 ⊢ (∃𝑗 ∈ ω Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → Pred(𝑅, 𝐴, 𝑌) ⊆ ∪ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
45	43, 44	syl 17	. . . . . 6 ⊢ ((suc 𝑖 ∈ ω ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ ∪ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
46		dftrpred2 31719	. . . . . 6 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
47	45, 46	syl6sseqr 3652	. . . . 5 ⊢ ((suc 𝑖 ∈ ω ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋))
48	4, 40, 47	syl2anc 693	. . . 4 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋))
49	48	ex 450	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))
50	49	rexlimdva 3031	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))
51	1, 50	syl5bi 232	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∪ ciun 4520   ↦ 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:  trpredelss  31732  frmin  31739