Theorem trpredlem1 30971

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 6982	. . 3 ⊢ (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
2		fr0g 7418	. . . . . 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 5604	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
4	2, 3	syl6eqss 3618	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐴)
5		fveq2 6103	. . . . . 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 3595	. . . . 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 235	. . . 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 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑋)
9		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑗
10		nfmpt1 4675	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎(𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦))
11	10, 8	nfrdg 7397	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
12		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎ω
13	11, 12	nfres 5319	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
14	13, 9	nffv 6110	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
15		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑒)
16	14, 15	nfiun 4484	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∪ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒)
17		predeq3 5601	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑒))
18	17	cbviunv 4495	. . . . . . . . . . . . 13 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒)
19	18	mpteq2i 4669	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑒))
20		rdgeq1 7394	. . . . . . . . . . . 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 5311	. . . . . . . . . . . 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 4470	. . . . . . . . . . 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 7420	. . . . . . . . . 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 4497	. . . . . . . . . . 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 5604	. . . . . . . . . . . 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 2911	. . . . . . . . . 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	syl6eqss 3618	. . . . . . . . 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 7421	. . . . . . . . . . 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 481	. . . . . . . . . 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 3924	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐴
33	31, 32	syl6eqss 3618	. . . . . . . . 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 828	. . . . . . . 8 ⊢ (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
35	34	adantr 480	. . . . . . 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 6103	. . . . . . . . 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 3595	. . . . . . . 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 481	. . . . . . 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 246	. . . . . 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 3010	. . . . 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 393	. . 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 6134	. . . 4 ⊢ (¬ 𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ∅)
45	44, 32	syl6eqss 3618	. . 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 175	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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455   ↦ cmpt 4643   ↾ cres 5040  Predcpred 5596  suc csuc 5642  ‘cfv 5804  ωcom 6957  reccrdg 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393

This theorem is referenced by:  trpredss  30973  trpredtr  30974  trpredmintr  30975  trpredrec  30982