Theorem trpredtr 9313

Description: Predecessors of a transitive predecessor are transitive predecessors. (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 9309	. 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 769	. . . . 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 7646	. . . . 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 488	. . . . . 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 3909	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)
7		predeq3 6144	. . . . . . . . 9 ⊢ (𝑡 = 𝑌 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑌))
8	7	sseq2d 3919	. . . . . . . 8 ⊢ (𝑡 = 𝑌 → (Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡) ↔ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)))
9	8	rspcev 3527	. . . . . . 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 4941	. . . . . . 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 589	. . . . 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 6708	. . . . . . 7 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∈ V
14		setlikespec 6161	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
15		trpredlem1 9310	. . . . . . . . . . . 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 3886	. . . . . . . . . 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 6161	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑡) ∈ V)
19	18	expcom 417	. . . . . . . . . . 11 ⊢ (𝑅 Se 𝐴 → (𝑡 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
20	19	adantl 485	. . . . . . . . . 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 3094	. . . . . . . 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 726	. . . . . . 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 7714	. . . . . . 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 590	. . . . . 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 2897	. . . . . . 7 ⊢ Ⅎ𝑓Pred(𝑅, 𝐴, 𝑋)
27		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑓𝑖
28		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑓∪ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡)
29		predeq3 6144	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑡))
30	29	cbviunv 4935	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑡 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑡)
31		iuneq1 4906	. . . . . . . . . 10 ⊢ (𝑎 = 𝑓 → ∪ 𝑡 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑡) = ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
32	30, 31	syl5eq 2783	. . . . . . . . 9 ⊢ (𝑎 = 𝑓 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
33	32	cbvmptv 5143	. . . . . . . 8 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ∈ V ↦ ∪ 𝑡 ∈ 𝑓 Pred(𝑅, 𝐴, 𝑡))
34		rdgeq1 8125	. . . . . . . 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 5830	. . . . . . . 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 4906	. . . . . . 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 8151	. . . . . 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 587	. . . . 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	sseqtrrd 3928	. . . 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 6695	. . . . . . . 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 3919	. . . . . . 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 3527	. . . . . 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 4941	. . . . . 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 9302	. . . . 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	sseqtrrdi 3938	. . . 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 587	. . 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 3196	. 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 245	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  Vcvv 3398   ⊆ wss 3853  ∪ ciun 4890   ↦ cmpt 5120   Se wse 5492   ↾ cres 5538  Predcpred 6139  suc csuc 6193  ‘cfv 6358  ωcom 7622  reccrdg 8123  TrPredctrpred 9300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-se 5495  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-trpred 9301

This theorem is referenced by:  trpredelss  9316  frmin  33459  frrlem16  33506