Theorem dfttc2g 37045

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 8412	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) = 𝐴)
2		rdgfnon 8403	. . . . . 6 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) Fn On
3		omsson 7864	. . . . . 6 ⊢ ω ⊆ On
4		peano1 7883	. . . . . 6 ⊢ ∅ ∈ ω
5		fnfvima 7231	. . . . . 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 1489	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)
7	1, 6	eqeltrrdi 2871	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
8		elssuni 4903	. . . 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 4876	. . . . . . . . . . 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 6894	. . . . . . . . . . . . . . 15 ⊢ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
14	13	uniex 7741	. . . . . . . . . . . . . 14 ⊢ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ∈ V
15		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) = rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
16		unieq 4882	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
17		unieq 4882	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∪ 𝑦 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
18	15, 16, 17	rdgsucmpt2 8415	. . . . . . . . . . . . . 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 597	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
20	19	eleq2d 2848	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ↔ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)))
21	20	biimpar 482	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧)) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
22	11, 21	sylan2 604	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
23		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧))
24	23	eleq2d 2848	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝑧 → (𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ↔ 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧)))
25	24	rspcev 3580	. . . . . . . . . 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 697	. . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ (𝑤 ∈ 𝑦 ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
27	26	an12s 661	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑧 ∈ ω ∧ 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦))
28	27	rexlimdvaa 3166	. . . . . . 7 ⊢ (𝑤 ∈ 𝑦 → (∃𝑧 ∈ ω 𝑦 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) → ∃𝑦 ∈ ω 𝑤 ∈ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦)))
29		rdgfun 8401	. . . . . . . 8 ⊢ Fun rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)
30		eluniima 7248	. . . . . . . 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 7248	. . . . . . . 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 411	. . . . 5 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
36	35	gen2 1825	. . . 4 ⊢ ∀𝑤∀𝑦((𝑤 ∈ 𝑦 ∧ 𝑦 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω)) → 𝑤 ∈ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
37		dftr2 5219	. . . 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 37035	. . 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 597	. 2 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
41		funiunfv 7246	. . . 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 6881	. . . . . . 7 ⊢ (𝑦 = ∅ → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅))
44	43	sseq1d 3967	. . . . . 6 ⊢ (𝑦 = ∅ → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴))
45		fveq2 6881	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) = (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧))
46	45	sseq1d 3967	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴))
47	23	sseq1d 3967	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴 ↔ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘suc 𝑧) ⊆ TC+ 𝐴))
48		ttcid 37031	. . . . . . 7 ⊢ 𝐴 ⊆ TC+ 𝐴
49	1, 48	eqsstrdi 3980	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘∅) ⊆ TC+ 𝐴)
50		uniss 4879	. . . . . . . . 9 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ ∪ TC+ 𝐴)
51		ttctr3 37034	. . . . . . . . 9 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
52	50, 51	sstrdi 3948	. . . . . . . 8 ⊢ ((rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑧) ⊆ TC+ 𝐴)
53	19	sseq1d 3967	. . . . . . . 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 412	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
58	57	iunssd 5014	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑦 ∈ ω (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴)‘𝑦) ⊆ TC+ 𝐴)
59	42, 58	eqsstrrid 3975	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω) ⊆ TC+ 𝐴)
60	40, 59	eqssd 3953	1 ⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191  Tr wtr 5217   “ cima 5663  Oncon0 6360  suc csuc 6362  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  ωcom 7860  reccrdg 8394  TC+ cttc 37025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-ttc 37026

This theorem is used by: (None)