Theorem trpredlem1 33179

Description: Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)

Assertion

Ref	Expression
trpredlem1	⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)

Distinct variable groups:   𝐴,𝑎,𝑦   𝑅,𝑎,𝑦   𝑋,𝑎

Allowed substitution hints:   𝐴(𝑖)   𝐵(𝑦,𝑖,𝑎)   𝑅(𝑖)   𝑋(𝑦,𝑖)

Proof of Theorem trpredlem1

Dummy variables 𝑒 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0suc 7586	. . 3 ⊢ (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
2		fr0g 8054	. . . . . 6 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
3		predss 6123	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
4	2, 3	eqsstrdi 3969	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐴)
5		fveq2 6645	. . . . . 6 ⊢ (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)‘∅))
6	5	sseq1d 3946	. . . . 5 ⊢ (𝑖 = ∅ → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐴))
7	4, 6	syl5ibr 249	. . . 4 ⊢ (𝑖 = ∅ → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
8		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑋)
9		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑗
10		nfmpt1 5128	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎(𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦))
11	10, 8	nfrdg 8033	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
12		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎ω
13	11, 12	nfres 5820	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
14	13, 9	nffv 6655	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
15		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑒)
16	14, 15	nfiun 4911	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒)
17		predeq3 6120	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑒))
18	17	cbviunv 4927	. . . . . . . . . . . . 13 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)
19	18	mpteq2i 5122	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒))
20		rdgeq1 8030	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)))
21		reseq1 5812	. . . . . . . . . . . 12 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)) → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
22	19, 20, 21	mp2b 10	. . . . . . . . . . 11 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
23		iuneq1 4897	. . . . . . . . . . 11 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒) = ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒))
24	8, 9, 16, 22, 23	frsucmpt 8056	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒))
25		iunss 4932	. . . . . . . . . . 11 ⊢ (∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴 ↔ ∀𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴)
26		predss 6123	. . . . . . . . . . . 12 ⊢ Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴)
28	25, 27	mprgbir 3121	. . . . . . . . . 10 ⊢ ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴
29	24, 28	eqsstrdi 3969	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
30	8, 9, 16, 22, 23	frsucmptn 8057	. . . . . . . . . . 11 ⊢ (¬ ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∅)
31	30	adantl 485	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ ¬ ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∅)
32		0ss 4304	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐴
33	31, 32	eqsstrdi 3969	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ ¬ ∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
34	29, 33	pm2.61dan 812	. . . . . . . 8 ⊢ (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
35	34	adantr 484	. . . . . . 7 ⊢ ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
36		fveq2 6645	. . . . . . . . 9 ⊢ (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))
37	36	sseq1d 3946	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴))
38	37	adantl 485	. . . . . . 7 ⊢ ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴))
39	35, 38	mpbird 260	. . . . . 6 ⊢ ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
40	39	rexlimiva 3240	. . . . 5 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
41	40	a1d 25	. . . 4 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
42	7, 41	jaoi 854	. . 3 ⊢ ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
43	1, 42	syl 17	. 2 ⊢ (𝑖 ∈ ω → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
44		nfvres 6681	. . . 4 ⊢ (¬ 𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ∅)
45	44, 32	eqsstrdi 3969	. . 3 ⊢ (¬ 𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
46	45	a1d 25	. 2 ⊢ (¬ 𝑖 ∈ ω → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
47	43, 46	pm2.61i 185	1 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∪ ciun 4881   ↦ cmpt 5110   ↾ cres 5521  Predcpred 6115  suc csuc 6161  ‘cfv 6324  ωcom 7560  reccrdg 8028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029

This theorem is referenced by:  trpredss  33181  trpredtr  33182  trpredmintr  33183  trpredrec  33190