Theorem ttctr 37459

Description: The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttctr	⊢ Tr TC+ 𝐴

Proof of Theorem ttctr

Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgfun 8452	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
2		eluniima 7277	. . . . . . . . 9 ⊢ (Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) → (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
4		peano2 7925	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
5		elunii 4875	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
6		nnon 7907	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
7		fvex 6921	. . . . . . . . . . . . . . . . 17 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
8	7	uniex 7782	. . . . . . . . . . . . . . . 16 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
9		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
10		unieq 4881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → ∪ 𝑤 = ∪ 𝑦)
11		unieq 4881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → ∪ 𝑤 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
12	9, 10, 11	rdgsucmpt2 8466	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ On ∧ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
13	6, 8, 12	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
14	13	eleq2d 2850	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) ↔ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)))
15	14	biimpar 483	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
16	5, 15	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
17		fveq2 6908	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
18	17	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ↔ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)))
19	18	rspcev 3580	. . . . . . . . . . . 12 ⊢ ((suc 𝑧 ∈ ω ∧ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)) → ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
20	4, 16, 19	syl2an2r 698	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
21		eluniima 7277	. . . . . . . . . . . 12 ⊢ (Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) → (𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤)))
22	1, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
23	20, 22	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
24	23	an12s 662	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑣 ∧ (𝑧 ∈ ω ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
25	24	rexlimdvaa 3168	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑣 → (∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
26	3, 25	biimtrid 245	. . . . . . 7 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
27	26	reximdv 3181	. . . . . 6 ⊢ (𝑢 ∈ 𝑣 → (∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → ∃𝑥 ∈ 𝐴 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
28		eliun 4958	. . . . . 6 ⊢ (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
29		eliun 4958	. . . . . 6 ⊢ (𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑥 ∈ 𝐴 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
30	27, 28, 29	3imtr4g 299	. . . . 5 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
31		df-ttc 37453	. . . . . 6 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
32	31	eleq2i 2856	. . . . 5 ⊢ (𝑣 ∈ TC+ 𝐴 ↔ 𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
33	31	eleq2i 2856	. . . . 5 ⊢ (𝑢 ∈ TC+ 𝐴 ↔ 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
34	30, 32, 33	3imtr4g 299	. . . 4 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ TC+ 𝐴 → 𝑢 ∈ TC+ 𝐴))
35	34	imp 412	. . 3 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
36	35	gen2 1829	. 2 ⊢ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
37		dftr2 5217	. 2 ⊢ (Tr TC+ 𝐴 ↔ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴))
38	36, 37	mpbir 234	1 ⊢ Tr TC+ 𝐴

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454  {csn 4588  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189  Tr wtr 5215   “ cima 5658  Oncon0 6380  suc csuc 6382  Fun wfun 6552  ‘cfv 6558  ωcom 7901  reccrdg 8445  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-iun 4956  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-ttc 37453

This theorem is used by:  ttctr2  37460  ttctr3  37461  ttcss  37464  ttcel  37466  ttcidm  37469  ttciunun  37477  ttcpwss  37481  dfttc3gw  37489  ttc0elw  37493  ttc0el  37501