Theorem dfttc3gw 37489

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 37500. (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 4930	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
2		treq 5222	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
3	2	ralab2 3658	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥 ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦))
4		simpr 490	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
5	3, 4	mpgbir 1832	. . . . 5 ⊢ ∀𝑥 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑥
6		trint 5233	. . . . 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 37462	. . . 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 705	. . 3 ⊢ TC+ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
10		df-tc 9763	. . . 4 ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
11		cleq1 15165	. . . . 5 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
12	11	adantl 487	. . . 4 ⊢ ((TC+ 𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)} = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
13		ttcexrg 37463	. . . 4 ⊢ (TC+ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
14		ttcid 37458	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ttctr 37459	. . . . . 6 ⊢ Tr TC+ 𝐴
16		sseq2 3939	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ TC+ 𝐴))
17		treq 5222	. . . . . . . 8 ⊢ (𝑦 = TC+ 𝐴 → (Tr 𝑦 ↔ Tr TC+ 𝐴))
18	16, 17	anbi12d 644	. . . . . . 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 711	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
21		intexab 5311	. . . . 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 7025	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
24	9, 23	sseqtrrid 3956	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ (TC‘𝐴))
25	14, 15	pm3.2i 476	. . . 4 ⊢ (𝐴 ⊆ TC+ 𝐴 ∧ Tr TC+ 𝐴)
26	18, 25	intmin3 4939	. . 3 ⊢ (TC+ 𝐴 ∈ 𝑉 → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ TC+ 𝐴)
27	23, 26	eqsstrd 3947	. 2 ⊢ (TC+ 𝐴 ∈ 𝑉 → (TC‘𝐴) ⊆ TC+ 𝐴)
28	24, 27	eqssd 3930	1 ⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∩ cint 4910  Tr wtr 5215  ‘cfv 6558  TCctc 9762  TC+ cttc 37452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-tc 9763  df-ttc 37453

This theorem is used by:  dfttc3g  37500