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Theorem trcl 9723

Description: For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9724 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

**Hypotheses**
Ref	Expression
trcl.1	⊢ 𝐴 ∈ V
trcl.2	⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
trcl.3	⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦)

**Assertion**
Ref	Expression
trcl	⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥))

Distinct variable groups: 𝑥,𝑧 𝑥,𝑦,𝐴 𝑥,𝐹,𝑦 Allowed substitution hints: 𝐴(𝑧) 𝐶(𝑥,𝑦,𝑧) 𝐹(𝑧)

Proof of Theorem trcl Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7879	. . . . 5 ⊢ ∅ ∈ ω
2		trcl.2	. . . . . . . 8 ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3	2	fveq1i 6893	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘∅)
4		trcl.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
5		fr0g 8436	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘∅) = 𝐴)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘∅) = 𝐴
7	3, 6	eqtr2i 2762	. . . . . 6 ⊢ 𝐴 = (𝐹‘∅)
8	7	eqimssi 4043	. . . . 5 ⊢ 𝐴 ⊆ (𝐹‘∅)
9		fveq2 6892	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
10	9	sseq2d 4015	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐴 ⊆ (𝐹‘𝑦) ↔ 𝐴 ⊆ (𝐹‘∅)))
11	10	rspcev 3613	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝐴 ⊆ (𝐹‘∅)) → ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦))
12	1, 8, 11	mp2an 691	. . . 4 ⊢ ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦)
13		ssiun 5050	. . . 4 ⊢ (∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦) → 𝐴 ⊆ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
14	12, 13	ax-mp 5	. . 3 ⊢ 𝐴 ⊆ ∪ 𝑦 ∈ ω (𝐹‘𝑦)
15		trcl.3	. . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦)
16	14, 15	sseqtrri 4020	. 2 ⊢ 𝐴 ⊆ 𝐶
17		dftr2 5268	. . . 4 ⊢ (Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∀𝑣∀𝑢((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)))
18		eliun 5002	. . . . . . . . 9 ⊢ (𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦))
19	18	anbi2i 624	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) ↔ (𝑣 ∈ 𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦)))
20		r19.42v 3191	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) ↔ (𝑣 ∈ 𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦)))
21	19, 20	bitr4i 278	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)))
22		elunii 4914	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → 𝑣 ∈ ∪ (𝐹‘𝑦))
23		ssun2 4174	. . . . . . . . . . 11 ⊢ ∪ (𝐹‘𝑦) ⊆ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦))
24		fvex 6905	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
25	24	uniex 7731	. . . . . . . . . . . . 13 ⊢ ∪ (𝐹‘𝑦) ∈ V
26	24, 25	unex 7733	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ∈ V
27		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
28		unieq 4920	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
29	27, 28	uneq12d 4165	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ ∪ 𝑥) = (𝑧 ∪ ∪ 𝑧))
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐹‘𝑦) → 𝑥 = (𝐹‘𝑦))
31		unieq 4920	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐹‘𝑦) → ∪ 𝑥 = ∪ (𝐹‘𝑦))
32	30, 31	uneq12d 4165	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥 ∪ ∪ 𝑥) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
33	2, 29, 32	frsucmpt2 8440	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ∈ V) → (𝐹‘suc 𝑦) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
34	26, 33	mpan2 690	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
35	23, 34	sseqtrrid 4036	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → ∪ (𝐹‘𝑦) ⊆ (𝐹‘suc 𝑦))
36	35	sseld 3982	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑣 ∈ ∪ (𝐹‘𝑦) → 𝑣 ∈ (𝐹‘suc 𝑦)))
37	22, 36	syl5 34	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → 𝑣 ∈ (𝐹‘suc 𝑦)))
38	37	reximia 3082	. . . . . . 7 ⊢ (∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
39	21, 38	sylbi 216	. . . . . 6 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
40		peano2 7881	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
41		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑢 = suc 𝑦 → (𝐹‘𝑢) = (𝐹‘suc 𝑦))
42	41	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝑢 = suc 𝑦 → (𝑣 ∈ (𝐹‘𝑢) ↔ 𝑣 ∈ (𝐹‘suc 𝑦)))
43	42	rspcev 3613	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑣 ∈ (𝐹‘suc 𝑦)) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
44	43	ex 414	. . . . . . . . . 10 ⊢ (suc 𝑦 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢)))
45	40, 44	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢)))
46	45	rexlimiv 3149	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
47		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
48	47	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑣 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑢)))
49	48	cbvrexvw 3236	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘𝑦) ↔ ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
50	46, 49	sylibr 233	. . . . . . 7 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘𝑦))
51		eliun 5002	. . . . . . 7 ⊢ (𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘𝑦))
52	50, 51	sylibr 233	. . . . . 6 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
53	39, 52	syl 17	. . . . 5 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
54	53	ax-gen 1798	. . . 4 ⊢ ∀𝑢((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
55	17, 54	mpgbir 1802	. . 3 ⊢ Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦)
56		treq 5274	. . . 4 ⊢ (𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦) → (Tr 𝐶 ↔ Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦)))
57	15, 56	ax-mp 5	. . 3 ⊢ (Tr 𝐶 ↔ Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦))
58	55, 57	mpbir 230	. 2 ⊢ Tr 𝐶
59		fveq2 6892	. . . . . . . 8 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
60	59	sseq1d 4014	. . . . . . 7 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘∅) ⊆ 𝑥))
61		fveq2 6892	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
62	61	sseq1d 4014	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘𝑦) ⊆ 𝑥))
63		fveq2 6892	. . . . . . . 8 ⊢ (𝑣 = suc 𝑦 → (𝐹‘𝑣) = (𝐹‘suc 𝑦))
64	63	sseq1d 4014	. . . . . . 7 ⊢ (𝑣 = suc 𝑦 → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘suc 𝑦) ⊆ 𝑥))
65	3, 6	eqtri 2761	. . . . . . . . . 10 ⊢ (𝐹‘∅) = 𝐴
66	65	sseq1i 4011	. . . . . . . . 9 ⊢ ((𝐹‘∅) ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥)
67	66	biimpri 227	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑥 → (𝐹‘∅) ⊆ 𝑥)
68	67	adantr 482	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐹‘∅) ⊆ 𝑥)
69		uniss 4917	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → ∪ (𝐹‘𝑦) ⊆ ∪ 𝑥)
70		df-tr 5267	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∪ 𝑥 ⊆ 𝑥)
71		sstr2 3990	. . . . . . . . . . . . . 14 ⊢ (∪ (𝐹‘𝑦) ⊆ ∪ 𝑥 → (∪ 𝑥 ⊆ 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
72	70, 71	biimtrid 241	. . . . . . . . . . . . 13 ⊢ (∪ (𝐹‘𝑦) ⊆ ∪ 𝑥 → (Tr 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
73	69, 72	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
74	73	anc2li 557	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹‘𝑦) ⊆ 𝑥 ∧ ∪ (𝐹‘𝑦) ⊆ 𝑥)))
75		unss 4185	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ⊆ 𝑥 ∧ ∪ (𝐹‘𝑦) ⊆ 𝑥) ↔ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥)
76	74, 75	imbitrdi 250	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥))
77	34	sseq1d 4014	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((𝐹‘suc 𝑦) ⊆ 𝑥 ↔ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥))
78	77	biimprd 247	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥))
79	76, 78	syl9r 78	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
80	79	com23 86	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (Tr 𝑥 → ((𝐹‘𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
81	80	adantld 492	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ((𝐹‘𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
82	60, 62, 64, 68, 81	finds2 7891	. . . . . 6 ⊢ (𝑣 ∈ ω → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐹‘𝑣) ⊆ 𝑥))
83	82	com12 32	. . . . 5 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝑣 ∈ ω → (𝐹‘𝑣) ⊆ 𝑥))
84	83	ralrimiv 3146	. . . 4 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
85		fveq2 6892	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
86	85	cbviunv 5044	. . . . . . 7 ⊢ ∪ 𝑦 ∈ ω (𝐹‘𝑦) = ∪ 𝑣 ∈ ω (𝐹‘𝑣)
87	15, 86	eqtri 2761	. . . . . 6 ⊢ 𝐶 = ∪ 𝑣 ∈ ω (𝐹‘𝑣)
88	87	sseq1i 4011	. . . . 5 ⊢ (𝐶 ⊆ 𝑥 ↔ ∪ 𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
89		iunss 5049	. . . . 5 ⊢ (∪ 𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
90	88, 89	bitri 275	. . . 4 ⊢ (𝐶 ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
91	84, 90	sylibr 233	. . 3 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)
92	91	ax-gen 1798	. 2 ⊢ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)
93	16, 58, 92	3pm3.2i 1340	1 ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∪ cun 3947 ⊆ wss 3949 ∅c0 4323 ∪ cuni 4909 ∪ ciun 4998 ↦ cmpt 5232 Tr wtr 5266 ↾ cres 5679 suc csuc 6367 ‘cfv 6544 ωcom 7855 reccrdg 8409

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410

This theorem is referenced by: tz9.1 9724 tz9.1c 9725

W3C validator