Theorem ttctr 37203

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 8402	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
2		eluniima 7242	. . . . . . . . 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 7884	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
5		elunii 4871	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
6		nnon 7866	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
7		fvex 6886	. . . . . . . . . . . . . . . . 17 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
8	7	uniex 7741	. . . . . . . . . . . . . . . 16 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
9		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
10		unieq 4877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → ∪ 𝑤 = ∪ 𝑦)
11		unieq 4877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → ∪ 𝑤 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
12	9, 10, 11	rdgsucmpt2 8416	. . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
18	17	eleq2d 2846	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ↔ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)))
19	18	rspcev 3576	. . . . . . . . . . . 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 7242	. . . . . . . . . . . 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 3164	. . . . . . . 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 3177	. . . . . 6 ⊢ (𝑢 ∈ 𝑣 → (∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → ∃𝑥 ∈ 𝐴 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
28		eliun 4954	. . . . . 6 ⊢ (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
29		eliun 4954	. . . . . 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 37197	. . . . . 6 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
32	31	eleq2i 2852	. . . . 5 ⊢ (𝑣 ∈ TC+ 𝐴 ↔ 𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
33	31	eleq2i 2852	. . . . 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 5213	. 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 2145  ∃wrex 3086  Vcvv 3450  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185  Tr wtr 5211   “ cima 5650  Oncon0 6351  suc csuc 6353  Fun wfun 6521  ‘cfv 6527  ωcom 7860  reccrdg 8395  TC+ cttc 37196

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 37197

This theorem is used by:  ttctr2  37204  ttctr3  37205  ttcss  37208  ttcel  37210  ttcidm  37213  ttciunun  37221  ttcpwss  37225  dfttc3gw  37233  ttc0elw  37237  ttc0el  37245