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Theorem trcl 9417

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 9418 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 7710	. . . . 5 ⊢ ∅ ∈ ω
2		trcl.2	. . . . . . . 8 ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3	2	fveq1i 6757	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘∅)
4		trcl.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
5		fr0g 8237	. . . . . . . 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 2767	. . . . . 6 ⊢ 𝐴 = (𝐹‘∅)
8	7	eqimssi 3975	. . . . 5 ⊢ 𝐴 ⊆ (𝐹‘∅)
9		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
10	9	sseq2d 3949	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐴 ⊆ (𝐹‘𝑦) ↔ 𝐴 ⊆ (𝐹‘∅)))
11	10	rspcev 3552	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝐴 ⊆ (𝐹‘∅)) → ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦))
12	1, 8, 11	mp2an 688	. . . 4 ⊢ ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹‘𝑦)
13		ssiun 4972	. . . 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 3954	. 2 ⊢ 𝐴 ⊆ 𝐶
17		dftr2 5189	. . . 4 ⊢ (Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∀𝑣∀𝑢((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)))
18		eliun 4925	. . . . . . . . 9 ⊢ (𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦) ↔ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦))
19	18	anbi2i 622	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) ↔ (𝑣 ∈ 𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦)))
20		r19.42v 3276	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) ↔ (𝑣 ∈ 𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹‘𝑦)))
21	19, 20	bitr4i 277	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)))
22		elunii 4841	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ (𝐹‘𝑦)) → 𝑣 ∈ ∪ (𝐹‘𝑦))
23		ssun2 4103	. . . . . . . . . . 11 ⊢ ∪ (𝐹‘𝑦) ⊆ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦))
24		fvex 6769	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
25	24	uniex 7572	. . . . . . . . . . . . 13 ⊢ ∪ (𝐹‘𝑦) ∈ V
26	24, 25	unex 7574	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ∈ V
27		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
28		unieq 4847	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
29	27, 28	uneq12d 4094	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ ∪ 𝑥) = (𝑧 ∪ ∪ 𝑧))
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐹‘𝑦) → 𝑥 = (𝐹‘𝑦))
31		unieq 4847	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐹‘𝑦) → ∪ 𝑥 = ∪ (𝐹‘𝑦))
32	30, 31	uneq12d 4094	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥 ∪ ∪ 𝑥) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
33	2, 29, 32	frsucmpt2 8241	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ∈ V) → (𝐹‘suc 𝑦) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
34	26, 33	mpan2 687	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)))
35	23, 34	sseqtrrid 3970	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → ∪ (𝐹‘𝑦) ⊆ (𝐹‘suc 𝑦))
36	35	sseld 3916	. . . . . . . . 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 3172	. . . . . . 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 7711	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
41		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑢 = suc 𝑦 → (𝐹‘𝑢) = (𝐹‘suc 𝑦))
42	41	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑢 = suc 𝑦 → (𝑣 ∈ (𝐹‘𝑢) ↔ 𝑣 ∈ (𝐹‘suc 𝑦)))
43	42	rspcev 3552	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑣 ∈ (𝐹‘suc 𝑦)) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
44	43	ex 412	. . . . . . . . . 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 3208	. . . . . . . 8 ⊢ (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹‘𝑢))
47		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
48	47	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑣 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑢)))
49	48	cbvrexvw 3373	. . . . . . . 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 4925	. . . . . . 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 1799	. . . 4 ⊢ ∀𝑢((𝑣 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦)) → 𝑣 ∈ ∪ 𝑦 ∈ ω (𝐹‘𝑦))
55	17, 54	mpgbir 1803	. . 3 ⊢ Tr ∪ 𝑦 ∈ ω (𝐹‘𝑦)
56		treq 5193	. . . 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 6756	. . . . . . . 8 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
60	59	sseq1d 3948	. . . . . . 7 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘∅) ⊆ 𝑥))
61		fveq2 6756	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
62	61	sseq1d 3948	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘𝑦) ⊆ 𝑥))
63		fveq2 6756	. . . . . . . 8 ⊢ (𝑣 = suc 𝑦 → (𝐹‘𝑣) = (𝐹‘suc 𝑦))
64	63	sseq1d 3948	. . . . . . 7 ⊢ (𝑣 = suc 𝑦 → ((𝐹‘𝑣) ⊆ 𝑥 ↔ (𝐹‘suc 𝑦) ⊆ 𝑥))
65	3, 6	eqtri 2766	. . . . . . . . . 10 ⊢ (𝐹‘∅) = 𝐴
66	65	sseq1i 3945	. . . . . . . . 9 ⊢ ((𝐹‘∅) ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥)
67	66	biimpri 227	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑥 → (𝐹‘∅) ⊆ 𝑥)
68	67	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐹‘∅) ⊆ 𝑥)
69		uniss 4844	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → ∪ (𝐹‘𝑦) ⊆ ∪ 𝑥)
70		df-tr 5188	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∪ 𝑥 ⊆ 𝑥)
71		sstr2 3924	. . . . . . . . . . . . . 14 ⊢ (∪ (𝐹‘𝑦) ⊆ ∪ 𝑥 → (∪ 𝑥 ⊆ 𝑥 → ∪ (𝐹‘𝑦) ⊆ 𝑥))
72	70, 71	syl5bi 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 555	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹‘𝑦) ⊆ 𝑥 ∧ ∪ (𝐹‘𝑦) ⊆ 𝑥)))
75		unss 4114	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ⊆ 𝑥 ∧ ∪ (𝐹‘𝑦) ⊆ 𝑥) ↔ ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥)
76	74, 75	syl6ib 250	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹‘𝑦) ∪ ∪ (𝐹‘𝑦)) ⊆ 𝑥))
77	34	sseq1d 3948	. . . . . . . . . . 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 490	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ((𝐹‘𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
82	60, 62, 64, 68, 81	finds2 7721	. . . . . 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 3106	. . . 4 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
85		fveq2 6756	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
86	85	cbviunv 4966	. . . . . . 7 ⊢ ∪ 𝑦 ∈ ω (𝐹‘𝑦) = ∪ 𝑣 ∈ ω (𝐹‘𝑣)
87	15, 86	eqtri 2766	. . . . . 6 ⊢ 𝐶 = ∪ 𝑣 ∈ ω (𝐹‘𝑣)
88	87	sseq1i 3945	. . . . 5 ⊢ (𝐶 ⊆ 𝑥 ↔ ∪ 𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
89		iunss 4971	. . . . 5 ⊢ (∪ 𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
90	88, 89	bitri 274	. . . 4 ⊢ (𝐶 ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹‘𝑣) ⊆ 𝑥)
91	84, 90	sylibr 233	. . 3 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)
92	91	ax-gen 1799	. 2 ⊢ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)
93	16, 58, 92	3pm3.2i 1337	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 395 ∧ w3a 1085 ∀wal 1537 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∪ cun 3881 ⊆ wss 3883 ∅c0 4253 ∪ cuni 4836 ∪ ciun 4921 ↦ cmpt 5153 Tr wtr 5187 ↾ cres 5582 suc csuc 6253 ‘cfv 6418 ωcom 7687 reccrdg 8211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212

This theorem is referenced by: tz9.1 9418 tz9.1c 9419

W3C validator