Theorem dfttc3gw 36821

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 36832. (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 4915	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
2		treq 5204	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
3	2	ralab2 3650	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥 ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦))
4		simpr 487	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
5	3, 4	mpgbir 1809	. . . . 5 ⊢ ∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥
6		trint 5215	. . . . 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 36794	. . . 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 700	. . 3 ⊢ TC+ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
10		df-tc 9676	. . . 4 ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
11		cleq1 14982	. . . . 5 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
12	11	adantl 484	. . . 4 ⊢ ((TC+ 𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
13		ttcexrg 36795	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
14		ttcid 36790	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ttctr 36791	. . . . . 6 ⊢ Tr TC+ 𝐴
16		sseq2 3953	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ TC+ 𝐴))
17		treq 5204	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (Tr 𝑦 ↔ Tr TC+ 𝐴))
18	16, 17	anbi12d 640	. . . . . . 7 ⊢ (𝑦 = TC+ 𝐴 → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)))
19	18	spcegv 3547	. . . . . 6 ⊢ (TC+ 𝐴 ∈ 𝑉 → ((𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴) → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
20	14, 15, 19	mp2ani 706	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
21		intexab 5292	. . . . 5 ⊢ (∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V)
22	20, 21	sylib 220	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V)
23	10, 12, 13, 22	fvmptd2 6969	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
24	9, 23	sseqtrrid 3970	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ (TC‘𝐴))
25	14, 15	pm3.2i 473	. . . 4 ⊢ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)
26	18, 25	intmin3 4924	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ TC+ 𝐴)
27	23, 26	eqsstrd 3961	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) ⊆ TC+ 𝐴)
28	24, 27	eqssd 3944	1 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132  {cab 2730  ∀wral 3066  Vcvv 3444   ⊆ wss 3895  ∩ cint 4895  Tr wtr 5197  ‘cfv 6506  TCctc 9675  TC+ cttc 36784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-iin 4942  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-tc 9676  df-ttc 36785

This theorem is referenced by:  dfttc3g  36832