Theorem dfttc3gw 36705

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 36716. (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 4909	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
2		treq 5199	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
3	2	ralab2 3643	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥 ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦))
4		simpr 484	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
5	3, 4	mpgbir 1801	. . . . 5 ⊢ ∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥
6		trint 5210	. . . . 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 36678	. . . 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 693	. . 3 ⊢ TC+ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
10		df-tc 9656	. . . 4 ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
11		cleq1 14945	. . . . 5 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
12	11	adantl 481	. . . 4 ⊢ ((TC+ 𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
13		ttcexrg 36679	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
14		ttcid 36674	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ttctr 36675	. . . . . 6 ⊢ Tr TC+ 𝐴
16		sseq2 3948	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ TC+ 𝐴))
17		treq 5199	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (Tr 𝑦 ↔ Tr TC+ 𝐴))
18	16, 17	anbi12d 633	. . . . . . 7 ⊢ (𝑦 = TC+ 𝐴 → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)))
19	18	spcegv 3539	. . . . . 6 ⊢ (TC+ 𝐴 ∈ 𝑉 → ((𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴) → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
20	14, 15, 19	mp2ani 699	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
21		intexab 5287	. . . . 5 ⊢ (∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V)
22	20, 21	sylib 218	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V)
23	10, 12, 13, 22	fvmptd2 6956	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
24	9, 23	sseqtrrid 3965	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ (TC‘𝐴))
25	14, 15	pm3.2i 470	. . . 4 ⊢ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)
26	18, 25	intmin3 4918	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ TC+ 𝐴)
27	23, 26	eqsstrd 3956	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) ⊆ TC+ 𝐴)
28	24, 27	eqssd 3939	1 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ∩ cint 4889  Tr wtr 5192  ‘cfv 6498  TCctc 9655  TC+ cttc 36668

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 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-tc 9656  df-ttc 36669

This theorem is referenced by:  dfttc3g  36716