Theorem ttctr 36858

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 8387	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
2		eluniima 7234	. . . . . . . . 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 7870	. . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
7		fvex 6880	. . . . . . . . . . . . . . . . 17 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
8	7	uniex 7724	. . . . . . . . . . . . . . . 16 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
9		eqid 2763	. . . . . . . . . . . . . . . . 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 8401	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
14	13	eleq2d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) ↔ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)))
15	14	biimpar 481	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
16	5, 15	sylan2 602	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
17		fveq2 6867	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
18	17	eleq2d 2849	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ↔ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)))
19	18	rspcev 3582	. . . . . . . . . . . 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 695	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
21		eluniima 7234	. . . . . . . . . . . 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 236	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
24	23	an12s 659	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑣 ∧ (𝑧 ∈ ω ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
25	24	rexlimdvaa 3165	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑣 → (∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
26	3, 25	biimtrid 244	. . . . . . 7 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
27	26	reximdv 3178	. . . . . 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 298	. . . . 5 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
31		df-ttc 36852	. . . . . 6 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
32	31	eleq2i 2855	. . . . 5 ⊢ (𝑣 ∈ TC+ 𝐴 ↔ 𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
33	31	eleq2i 2855	. . . . 5 ⊢ (𝑢 ∈ TC+ 𝐴 ↔ 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
34	30, 32, 33	3imtr4g 298	. . . 4 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ TC+ 𝐴 → 𝑢 ∈ TC+ 𝐴))
35	34	imp 410	. . 3 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
36	35	gen2 1817	. 2 ⊢ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
37		dftr2 5210	. 2 ⊢ (Tr TC+ 𝐴 ↔ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴))
38	36, 37	mpbir 233	1 ⊢ Tr TC+ 𝐴

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1559   = wceq 1561   ∈ wcel 2143  ∃wrex 3087  Vcvv 3455  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5182  Tr wtr 5208   “ cima 5651  Oncon0 6346  suc csuc 6348  Fun wfun 6515  ‘cfv 6521  ωcom 7846  reccrdg 8380  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-ttc 36852

This theorem is referenced by:  ttctr2  36859  ttctr3  36860  ttcss  36863  ttcel  36865  ttcidm  36868  ttciunun  36876  ttcpwss  36880  dfttc3gw  36888  ttc0elw  36892  ttc0el  36900