Theorem trpredtr 32678

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 32674	. 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 765	. . . . 5 ⊢ ((((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 7458	. . . . 5 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
4	2, 3	syl 17	. . . 4 ⊢ ((((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 485	. . . . . 6 ⊢ ((((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 3910	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)
7		predeq3 6027	. . . . . . . . 9 ⊢ (𝑡 = 𝑌 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑌))
8	7	sseq2d 3920	. . . . . . . 8 ⊢ (𝑡 = 𝑌 → (Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡) ↔ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)))
9	8	rspcev 3559	. . . . . . 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)) → ∃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 4869	. . . . . . 7 ⊢ (∃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	. . . . . 6 ⊢ ((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 586	. . . . 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) ⊆ ∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
13		fvex 6551	. . . . . . 7 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∈ V
14		setlikespec 6044	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
15		trpredlem1 32675	. . . . . . . . . . . 12 ⊢ (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	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
17	16	sseld 3888	. . . . . . . . . 10 ⊢ ((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 6044	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑡) ∈ V)
19	18	expcom 414	. . . . . . . . . . 11 ⊢ (𝑅 Se 𝐴 → (𝑡 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
20	19	adantl 482	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑡 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
21	17, 20	syld 47	. . . . . . . . 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))
22	21	ralrimiv 3148	. . . . . . . 8 ⊢ ((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 722	. . . . . . 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)
24		iunexg 7520	. . . . . . 7 ⊢ ((((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 587	. . . . . 6 ⊢ ((((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 2949	. . . . . . 7 ⊢ Ⅎ𝑓Pred(𝑅, 𝐴, 𝑋)
27		nfcv 2949	. . . . . . 7 ⊢ Ⅎ𝑓𝑖
28		nfcv 2949	. . . . . . 7 ⊢ Ⅎ𝑓∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡)
29		predeq3 6027	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑡))
30	29	cbviunv 4866	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑡 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑡)
31		iuneq1 4840	. . . . . . . . . 10 ⊢ (𝑎 = 𝑓 → ∪ 𝑡 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
32	30, 31	syl5eq 2843	. . . . . . . . 9 ⊢ (𝑎 = 𝑓 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
33	32	cbvmptv 5061	. . . . . . . 8 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
34		rdgeq1 7899	. . . . . . . 8 ⊢ ((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 5728	. . . . . . . 8 ⊢ (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	. . . . . . 7 ⊢ (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 4840	. . . . . . 7 ⊢ (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 7925	. . . . . 6 ⊢ ((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 584	. . . . 5 ⊢ ((((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 3929	. . . 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) ⊆ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖))
41		fveq2 6538	. . . . . . . 8 ⊢ (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 3920	. . . . . . 7 ⊢ (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 3559	. . . . . 6 ⊢ ((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 4869	. . . . . 6 ⊢ (∃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	. . . . 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) ⊆ ∪ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
46		dftrpred2 32667	. . . . 5 ⊢ 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 3939	. . . 4 ⊢ ((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 584	. . 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))
49	48	rexlimdva2 3250	. 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)))
50	1, 49	syl5bi 243	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  Vcvv 3437   ⊆ wss 3859  ∪ ciun 4825   ↦ cmpt 5041   Se wse 5400   ↾ cres 5445  Predcpred 6022  suc csuc 6068  ‘cfv 6225  ωcom 7436  reccrdg 7897  TrPredctrpred 32665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-trpred 32666

This theorem is referenced by:  trpredelss  32680  frmin  32693  frrlem16  32752