Theorem ttctr 36734

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 8349	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
2		eluniima 7197	. . . . . . . . 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 7833	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
5		elunii 4845	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
6		nnon 7815	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
7		fvex 6843	. . . . . . . . . . . . . . . . 17 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
8	7	uniex 7687	. . . . . . . . . . . . . . . 16 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
9		eqid 2741	. . . . . . . . . . . . . . . . 17 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
10		unieq 4851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → ∪ 𝑤 = ∪ 𝑦)
11		unieq 4851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → ∪ 𝑤 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
12	9, 10, 11	rdgsucmpt2 8363	. . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
14	13	eleq2d 2827	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) ↔ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)))
15	14	biimpar 479	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
16	5, 15	sylan2 600	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
17		fveq2 6830	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
18	17	eleq2d 2827	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ↔ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)))
19	18	rspcev 3561	. . . . . . . . . . . 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 692	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
21		eluniima 7197	. . . . . . . . . . . 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 656	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑣 ∧ (𝑧 ∈ ω ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
25	24	rexlimdvaa 3143	. . . . . . . 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 3156	. . . . . 6 ⊢ (𝑢 ∈ 𝑣 → (∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → ∃𝑥 ∈ 𝐴 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
28		eliun 4927	. . . . . 6 ⊢ (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
29		eliun 4927	. . . . . 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 36728	. . . . . 6 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
32	31	eleq2i 2833	. . . . 5 ⊢ (𝑣 ∈ TC+ 𝐴 ↔ 𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
33	31	eleq2i 2833	. . . . 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 408	. . 3 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
36	35	gen2 1804	. 2 ⊢ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
37		dftr2 5183	. 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 397  ∀wal 1546   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  Vcvv 3433  {csn 4557  ∪ cuni 4840  ∪ ciun 4923   ↦ cmpt 5155  Tr wtr 5181   “ cima 5623  Oncon0 6313  suc csuc 6315  Fun wfun 6482  ‘cfv 6488  ωcom 7809  reccrdg 8342  TC+ cttc 36727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-ttc 36728

This theorem is referenced by:  ttctr2  36735  ttctr3  36736  ttcss  36739  ttcel  36741  ttcidm  36744  ttciunun  36752  ttcpwss  36756  dfttc3gw  36764  ttc0elw  36768  ttc0el  36776