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Theorem trcl 9698

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 9699 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 7886	. . . . 5 ⊢ ∅ ∈ ω
2		trcl.2	. . . . . . . 8 ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3	2	fveq1i 6884	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘∅)
4		trcl.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
5		fr0g 8424	. . . . . . . 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 2787	. . . . . 6 ⊢ 𝐴 = (𝐹‘∅)
8	7	eqimssi 3998	. . . . 5 ⊢ 𝐴 ⊆ (𝐹‘∅)
9		fveq2 6883	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
10	9	sseq2d 3970	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐴 ⊆ (𝐹‘𝑦) ↔ 𝐴 ⊆ (𝐹‘∅)))
11	10	rspcev 3582	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝐴 ⊆ (𝐹‘∅)) → ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦))
12	1, 8, 11	mp2an 704	. . . 4 ⊢ ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦)
13		ssiun 5012	. . . 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 3987	. 2 ⊢ 𝐴 ⊆ 𝐶
17		dftr2 5221	. . . 4 ⊢ (Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∀𝑣∀𝑢((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)))
18		eliun 4961	. . . . . . . . 9 ⊢ (𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦))
19	18	anbi2i 634	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) ↔ (𝑣 ∈ 𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦)))
20		r19.42v 3197	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) ↔ (𝑣 ∈ 𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦)))
21	19, 20	bitr4i 281	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)))
22		elunii 4878	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → 𝑣 ∈ ∪ (𝐹‘𝑦))
23		ssun2 4133	. . . . . . . . . . 11 ⊢ ∪ (𝐹‘𝑦) ⊆ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦))
24		fvex 6896	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
25	24	uniex 7741	. . . . . . . . . . . . 13 ⊢ ∪ (𝐹‘𝑦) ∈ V
26	24, 25	unex 7744	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ∈ V
27		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
28		unieq 4884	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
29	27, 28	uneq12d 4124	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ ∪ 𝑥) = (𝑧 ∪ ∪ 𝑧))
30		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐹‘𝑦) → 𝑥 = (𝐹‘𝑦))
31		unieq 4884	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐹‘𝑦) → ∪ 𝑥 = ∪ (𝐹‘𝑦))
32	30, 31	uneq12d 4124	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥 ∪ ∪ 𝑥) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
33	2, 29, 32	frsucmpt2 8428	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ∈ V) → (𝐹‘suc 𝑦) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
34	26, 33	mpan2 703	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
35	23, 34	sseqtrrid 3981	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → ∪ (𝐹‘𝑦) ⊆ (𝐹‘suc 𝑦))
36	35	sseld 3937	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑣 ∈ ∪ (𝐹‘𝑦) → 𝑣 ∈ (𝐹‘suc 𝑦)))
37	22, 36	syl5 35	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → 𝑣 ∈ (𝐹‘suc 𝑦)))
38	37	reximia 3100	. . . . . . 7 ⊢ (∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
39	21, 38	sylbi 220	. . . . . 6 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
40		peano2 7887	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
41		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑢 = suc 𝑦 → (𝐹‘𝑢) = (𝐹‘suc 𝑦))
42	41	eleq2d 2849	. . . . . . . . . . . 12 ⊢ (𝑢 = suc 𝑦 → (𝑣 ∈ (𝐹‘𝑢) ↔ 𝑣 ∈ (𝐹‘suc 𝑦)))
43	42	rspcev 3582	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑣 ∈ (𝐹‘suc 𝑦)) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
44	43	ex 417	. . . . . . . . . 10 ⊢ (suc 𝑦 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢)))
45	40, 44	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢)))
46	45	rexlimiv 3159	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
47		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
48	47	eleq2d 2849	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑣 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑢)))
49	48	cbvrexvw 3244	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘𝑦) ↔ ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
50	46, 49	sylibr 237	. . . . . . 7 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘𝑦))
51		eliun 4961	. . . . . . 7 ⊢ (𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘𝑦))
52	50, 51	sylibr 237	. . . . . 6 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
53	39, 52	syl 18	. . . . 5 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
54	53	ax-gen 1825	. . . 4 ⊢ ∀𝑢((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
55	17, 54	mpgbir 1829	. . 3 ⊢ Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦)
56		treq 5226	. . . 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 234	. 2 ⊢ Tr 𝐶
59		fveq2 6883	. . . . . . . 8 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
60	59	sseq1d 3969	. . . . . . 7 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘∅) ⊆ 𝑥))
61		fveq2 6883	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
62	61	sseq1d 3969	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘𝑦) ⊆ 𝑥))
63		fveq2 6883	. . . . . . . 8 ⊢ (𝑣 = suc 𝑦 → (𝐹‘𝑣) = (𝐹‘suc 𝑦))
64	63	sseq1d 3969	. . . . . . 7 ⊢ (𝑣 = suc 𝑦 → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘suc 𝑦) ⊆ 𝑥))
65	3, 6	eqtri 2786	. . . . . . . . 9 ⊢ (𝐹‘∅) = 𝐴
66	65	sseq1i 3966	. . . . . . . 8 ⊢ ((𝐹‘∅) ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥)
67	66	biranri 510	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐹‘∅) ⊆ 𝑥)
68		uniss 4881	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → ∪ (𝐹‘𝑦) ⊆ ∪ 𝑥)
69		df-tr 5220	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∪ 𝑥 ⊆ 𝑥)
70		sstr2 3945	. . . . . . . . . . . . . 14 ⊢ (∪ (𝐹‘𝑦) ⊆ ∪ 𝑥 → (∪ 𝑥 ⊆ 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
71	69, 70	biimtrid 245	. . . . . . . . . . . . 13 ⊢ (∪ (𝐹‘𝑦) ⊆ ∪ 𝑥 → (Tr 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
72	68, 71	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
73	72	anc2li 564	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹‘𝑦) ⊆ 𝑥 ∧ ∪ (𝐹‘𝑦) ⊆ 𝑥)))
74		unss 4144	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ⊆ 𝑥 ∧ ∪ (𝐹‘𝑦) ⊆ 𝑥) ↔ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥)
75	73, 74	imbitrdi 254	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥))
76	34	sseq1d 3969	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((𝐹‘suc 𝑦) ⊆ 𝑥 ↔ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥))
77	76	biimprd 251	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥))
78	75, 77	syl9r 79	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
79	78	com23 87	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (Tr 𝑥 → ((𝐹‘𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
80	79	adantld 495	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ((𝐹‘𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
81	60, 62, 64, 67, 80	finds2 7896	. . . . . 6 ⊢ (𝑣 ∈ ω → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐹‘𝑣) ⊆ 𝑥))
82	81	com12 33	. . . . 5 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝑣 ∈ ω → (𝐹‘𝑣) ⊆ 𝑥))
83	82	ralrimiv 3156	. . . 4 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
84		fveq2 6883	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
85	84	cbviunv 5004	. . . . . . 7 ⊢ ∪ 𝑦 ∈ ω (𝐹‘𝑦) = ∪ 𝑣 ∈ ω (𝐹‘𝑣)
86	15, 85	eqtri 2786	. . . . . 6 ⊢ 𝐶 = ∪ 𝑣 ∈ ω (𝐹‘𝑣)
87	86	sseq1i 3966	. . . . 5 ⊢ (𝐶 ⊆ 𝑥 ↔ ∪ 𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
88		iunss 5010	. . . . 5 ⊢ (∪ 𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
89	87, 88	bitri 278	. . . 4 ⊢ (𝐶 ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
90	83, 89	sylibr 237	. . 3 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)
91	90	ax-gen 1825	. 2 ⊢ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)
92	16, 58, 91	3pm3.2i 1358	1 ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∪ cun 3904 ⊆ wss 3906 ∅c0 4287 ∪ cuni 4873 ∪ ciun 4957 ↦ cmpt 5193 Tr wtr 5219 ↾ cres 5665 suc csuc 6364 ‘cfv 6538 ωcom 7863 reccrdg 8397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-om 7864 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398

This theorem is referenced by: tz9.1 9699 tz9.1c 9700

W3C validator