Theorem dfttc2g 37216

Description: A shorter expression for the transitive closure of a set. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
dfttc2g	⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))

Proof of Theorem dfttc2g

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdg0g 8413	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) = 𝐴)
2		rdgfnon 8404	. . . . . 6 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) Fn On
3		omsson 7864	. . . . . 6 ⊢ ω ⊆ On
4		peano1 7883	. . . . . 6 ⊢ ∅ ∈ ω
5		fnfvima 7227	. . . . . 6 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) Fn On ∧ ω ⊆ On ∧ ∅ ∈ ω) → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
6	2, 3, 4, 5	mp3an 1490	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
7	1, 6	eqeltrrdi 2869	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
8		elssuni 4898	. . . 4 ⊢ (𝐴 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) → 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
9	7, 8	syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
10		peano2 7884	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
11		elunii 4871	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
12		nnon 7866	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
13		fvex 6886	. . . . . . . . . . . . . . 15 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
14	13	uniex 7741	. . . . . . . . . . . . . 14 ⊢ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
15		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) = rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
16		unieq 4877	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
17		unieq 4877	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∪ 𝑦 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
18	15, 16, 17	rdgsucmpt2 8416	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ On ∧ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V) → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
19	12, 14, 18	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
20	19	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ↔ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)))
21	20	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
22	11, 21	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
23		fveq2 6873	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
24	23	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝑧 → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ↔ 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧)))
25	24	rspcev 3576	. . . . . . . . . 10 ⊢ ((suc 𝑧 ∈ ω ∧ 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧)) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
26	10, 22, 25	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
27	26	an12s 662	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑧 ∈ ω ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
28	27	rexlimdvaa 3164	. . . . . . 7 ⊢ (𝑤 ∈ 𝑦 → (∃𝑧 ∈ ω 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦)))
29		rdgfun 8402	. . . . . . . 8 ⊢ Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
30		eluniima 7242	. . . . . . . 8 ⊢ (Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) → (𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ↔ ∃𝑧 ∈ ω 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)))
31	29, 30	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ↔ ∃𝑧 ∈ ω 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
32		eluniima 7242	. . . . . . . 8 ⊢ (Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) → (𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ↔ ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦)))
33	29, 32	ax-mp 5	. . . . . . 7 ⊢ (𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ↔ ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
34	28, 31, 33	3imtr4g 299	. . . . . 6 ⊢ (𝑤 ∈ 𝑦 → (𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)))
35	34	imp 412	. . . . 5 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
36	35	gen2 1829	. . . 4 ⊢ ∀𝑤∀𝑦((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
37		dftr2 5213	. . . 4 ⊢ (Tr ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ↔ ∀𝑤∀𝑦((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)))
38	36, 37	mpbir 234	. . 3 ⊢ Tr ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
39		ttcmin 37206	. . 3 ⊢ ((𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ∧ Tr ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → TC+ 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
40	9, 38, 39	sylancl 598	. 2 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
41		funiunfv 7240	. . . 4 ⊢ (Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) → ∪ 𝑦 ∈ ω (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
42	29, 41	ax-mp 5	. . 3 ⊢ ∪ 𝑦 ∈ ω (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
43		fveq2 6873	. . . . . . 7 ⊢ (𝑦 = ∅ → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅))
44	43	sseq1d 3961	. . . . . 6 ⊢ (𝑦 = ∅ → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴))
45		fveq2 6873	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
46	45	sseq1d 3961	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
47	23	sseq1d 3961	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴))
48		ttcid 37202	. . . . . . 7 ⊢ 𝐴 ⊆ TC+ 𝐴
49	1, 48	eqsstrdi 3974	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴)
50		uniss 4874	. . . . . . . . 9 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ ∪ TC+ 𝐴)
51		ttctr3 37205	. . . . . . . . 9 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
52	50, 51	sstrdi 3942	. . . . . . . 8 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴)
53	19	sseq1d 3961	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴 ↔ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
54	52, 53	imbitrrid 249	. . . . . . 7 ⊢ (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴))
55	54	a1d 26	. . . . . 6 ⊢ (𝑧 ∈ ω → (𝐴 ∈ 𝑉 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴)))
56	44, 46, 47, 49, 55	finds2 7893	. . . . 5 ⊢ (𝑦 ∈ ω → (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴))
57	56	impcom 413	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
58	57	iunssd 5008	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑦 ∈ ω (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
59	42, 58	eqsstrrid 3969	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ⊆ TC+ 𝐴)
60	40, 59	eqssd 3947	1 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185  Tr wtr 5211   “ cima 5650  Oncon0 6351  suc csuc 6353  Fun wfun 6521   Fn wfn 6522  ‘cfv 6527  ωcom 7860  reccrdg 8395  TC+ cttc 37196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 37197

This theorem is used by: (None)