Theorem dfttc2g 36871

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 8398	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) = 𝐴)
2		rdgfnon 8389	. . . . . 6 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) Fn On
3		omsson 7850	. . . . . 6 ⊢ ω ⊆ On
4		peano1 7869	. . . . . 6 ⊢ ∅ ∈ ω
5		fnfvima 7217	. . . . . 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 1483	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
7	1, 6	eqeltrrdi 2872	. . . 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 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
10		peano2 7870	. . . . . . . . . 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 7852	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
13		fvex 6880	. . . . . . . . . . . . . . 15 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
14	13	uniex 7724	. . . . . . . . . . . . . 14 ⊢ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
15		eqid 2763	. . . . . . . . . . . . . . 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 8401	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
20	19	eleq2d 2849	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ↔ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)))
21	20	biimpar 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
22	11, 21	sylan2 602	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
23		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
24	23	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝑧 → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ↔ 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧)))
25	24	rspcev 3582	. . . . . . . . . 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 695	. . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
27	26	an12s 659	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑧 ∈ ω ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
28	27	rexlimdvaa 3165	. . . . . . 7 ⊢ (𝑤 ∈ 𝑦 → (∃𝑧 ∈ ω 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦)))
29		rdgfun 8387	. . . . . . . 8 ⊢ Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
30		eluniima 7234	. . . . . . . 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 7234	. . . . . . . 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 298	. . . . . 6 ⊢ (𝑤 ∈ 𝑦 → (𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)))
35	34	imp 410	. . . . 5 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
36	35	gen2 1817	. . . 4 ⊢ ∀𝑤∀𝑦((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
37		dftr2 5210	. . . 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 233	. . 3 ⊢ Tr ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
39		ttcmin 36861	. . 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 595	. 2 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
41		funiunfv 7232	. . . 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 6867	. . . . . . 7 ⊢ (𝑦 = ∅ → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅))
44	43	sseq1d 3968	. . . . . 6 ⊢ (𝑦 = ∅ → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴))
45		fveq2 6867	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
46	45	sseq1d 3968	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
47	23	sseq1d 3968	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴))
48		ttcid 36857	. . . . . . 7 ⊢ 𝐴 ⊆ TC+ 𝐴
49	1, 48	eqsstrdi 3981	. . . . . 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 36860	. . . . . . . . 9 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
52	50, 51	sstrdi 3949	. . . . . . . 8 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴)
53	19	sseq1d 3968	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴 ↔ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
54	52, 53	imbitrrid 248	. . . . . . 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 7879	. . . . 5 ⊢ (𝑦 ∈ ω → (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴))
57	56	impcom 411	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
58	57	iunssd 5009	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑦 ∈ ω (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
59	42, 58	eqsstrrid 3976	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ⊆ TC+ 𝐴)
60	40, 59	eqssd 3954	1 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1559   = wceq 1561   ∈ wcel 2143  ∃wrex 3087  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5182  Tr wtr 5208   “ cima 5651  Oncon0 6346  suc csuc 6348  Fun wfun 6515   Fn wfn 6516  ‘cfv 6521  ωcom 7846  reccrdg 8380  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-ttc 36852

This theorem is referenced by: (None)