Theorem dfttc3gw 37062

Description: If the transitive closure of 𝐴 is a set, then its value is (TC‘𝐴). If we assume Transitive Containment, then we can weaken the hypothesis to 𝐴 ∈ 𝑉, see dfttc3g 37073. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
dfttc3gw	⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

Proof of Theorem dfttc3gw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssmin 4931	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
2		treq 5224	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
3	2	ralab2 3659	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥 ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦))
4		simpr 489	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
5	3, 4	mpgbir 1828	. . . . 5 ⊢ ∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥
6		trint 5235	. . . . 5 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥 → Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
7	5, 6	ax-mp 5	. . . 4 ⊢ Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
8		ttcmin 37035	. . . 4 ⊢ ((𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∧ Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}) → TC+ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
9	1, 7, 8	mp2an 704	. . 3 ⊢ TC+ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
10		df-tc 9702	. . . 4 ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
11		cleq1 15027	. . . . 5 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
12	11	adantl 486	. . . 4 ⊢ ((TC+ 𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
13		ttcexrg 37036	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
14		ttcid 37031	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ttctr 37032	. . . . . 6 ⊢ Tr TC+ 𝐴
16		sseq2 3962	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ TC+ 𝐴))
17		treq 5224	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (Tr 𝑦 ↔ Tr TC+ 𝐴))
18	16, 17	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = TC+ 𝐴 → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)))
19	18	spcegv 3555	. . . . . 6 ⊢ (TC+ 𝐴 ∈ 𝑉 → ((𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴) → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
20	14, 15, 19	mp2ani 710	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
21		intexab 5315	. . . . 5 ⊢ (∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V)
22	20, 21	sylib 221	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V)
23	10, 12, 13, 22	fvmptd2 6998	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
24	9, 23	sseqtrrid 3979	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ (TC‘𝐴))
25	14, 15	pm3.2i 475	. . . 4 ⊢ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)
26	18, 25	intmin3 4940	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ TC+ 𝐴)
27	23, 26	eqsstrd 3970	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) ⊆ TC+ 𝐴)
28	24, 27	eqssd 3953	1 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  ∀wral 3078  Vcvv 3454   ⊆ wss 3904  ∩ cint 4911  Tr wtr 5217  ‘cfv 6536  TCctc 9701  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-int 4912  df-iun 4957  df-iin 4958  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-tc 9702  df-ttc 37026

This theorem is used by:  dfttc3g  37073