Theorem dfttc2g 36676

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 8355	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) = 𝐴)
2		rdgfnon 8346	. . . . . 6 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) Fn On
3		omsson 7810	. . . . . 6 ⊢ ω ⊆ On
4		peano1 7829	. . . . . 6 ⊢ ∅ ∈ ω
5		fnfvima 7177	. . . . . 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 1464	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
7	1, 6	eqeltrrdi 2844	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
8		elssuni 4871	. . . 4 ⊢ (𝐴 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) → 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
9	7, 8	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
10		peano2 7830	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
11		elunii 4845	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
12		nnon 7812	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
13		fvex 6842	. . . . . . . . . . . . . . 15 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
14	13	uniex 7684	. . . . . . . . . . . . . 14 ⊢ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
15		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) = rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
16		unieq 4851	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
17		unieq 4851	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∪ 𝑦 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
18	15, 16, 17	rdgsucmpt2 8358	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
20	19	eleq2d 2821	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ↔ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)))
21	20	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
22	11, 21	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
23		fveq2 6829	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
24	23	eleq2d 2821	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝑧 → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ↔ 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧)))
25	24	rspcev 3562	. . . . . . . . . 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 686	. . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
27	26	an12s 650	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑧 ∈ ω ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
28	27	rexlimdvaa 3137	. . . . . . 7 ⊢ (𝑤 ∈ 𝑦 → (∃𝑧 ∈ ω 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦)))
29		rdgfun 8344	. . . . . . . 8 ⊢ Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
30		eluniima 7194	. . . . . . . 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 7194	. . . . . . . 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 296	. . . . . 6 ⊢ (𝑤 ∈ 𝑦 → (𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)))
35	34	imp 406	. . . . 5 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
36	35	gen2 1798	. . . 4 ⊢ ∀𝑤∀𝑦((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
37		dftr2 5183	. . . 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 231	. . 3 ⊢ Tr ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
39		ttcmin 36666	. . 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 587	. 2 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
41		funiunfv 7192	. . . 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 6829	. . . . . . 7 ⊢ (𝑦 = ∅ → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅))
44	43	sseq1d 3948	. . . . . 6 ⊢ (𝑦 = ∅ → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴))
45		fveq2 6829	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
46	45	sseq1d 3948	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
47	23	sseq1d 3948	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴))
48		ttcid 36662	. . . . . . 7 ⊢ 𝐴 ⊆ TC+ 𝐴
49	1, 48	eqsstrdi 3961	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴)
50		uniss 4848	. . . . . . . . 9 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ ∪ TC+ 𝐴)
51		ttctr3 36665	. . . . . . . . 9 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
52	50, 51	sstrdi 3929	. . . . . . . 8 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴)
53	19	sseq1d 3948	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴 ↔ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
54	52, 53	imbitrrid 246	. . . . . . 7 ⊢ (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴))
55	54	a1d 25	. . . . . 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 7838	. . . . 5 ⊢ (𝑦 ∈ ω → (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴))
57	56	impcom 407	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
58	57	iunssd 4982	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑦 ∈ ω (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
59	42, 58	eqsstrrid 3956	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ⊆ TC+ 𝐴)
60	40, 59	eqssd 3934	1 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃wrex 3059  Vcvv 3427   ⊆ wss 3885  ∅c0 4263  ∪ cuni 4840  ∪ ciun 4923   ↦ cmpt 5155  Tr wtr 5181   “ cima 5623  Oncon0 6312  suc csuc 6314  Fun wfun 6481   Fn wfn 6482  ‘cfv 6487  ωcom 7806  reccrdg 8337  TC+ cttc 36656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-ttc 36657

This theorem is referenced by: (None)