Theorem ttcmin 36861

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

Assertion

Ref	Expression
ttcmin	⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

Proof of Theorem ttcmin

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

Step	Hyp	Ref	Expression
1		df-ttc 36852	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
2		ssel2 3932	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3		rdgfun 8387	. . . . . . 7 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
4		funiunfv 7232	. . . . . . 7 ⊢ (Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) → ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
6		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅))
7	6	sseq1d 3968	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵))
8		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
9	8	sseq1d 3968	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵))
10		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤))
11	10	sseq1d 3968	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵))
12		vsnex 5393	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ V
13	12	rdg0 8392	. . . . . . . . . . 11 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) = {𝑥}
14		snssi 4745	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
15	13, 14	eqsstrid 3975	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
16	15	adantr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
17		nnon 7852	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
18		fvex 6880	. . . . . . . . . . . . . 14 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
19	18	uniex 7724	. . . . . . . . . . . . 13 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
20		eqid 2763	. . . . . . . . . . . . . 14 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
21		unieq 4877	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦)
22		unieq 4877	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) → ∪ 𝑧 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
23	20, 21, 22	rdgsucmpt2 8401	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ On ∧ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
24	17, 19, 23	sylancl 595	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
25	24	3ad2ant1 1147	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
26		uniss 4874	. . . . . . . . . . . . 13 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
27	26	3ad2ant3 1149	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
28		simp2r 1215	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → Tr 𝐵)
29		df-tr 5209	. . . . . . . . . . . . 13 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵)
30	28, 29	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ 𝐵 ⊆ 𝐵)
31	27, 30	sstrd 3947	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵)
32	25, 31	eqsstrd 3971	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵)
33	32	3exp 1133	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵)))
34	7, 9, 11, 16, 33	finds2 7879	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵))
35	34	impcom 411	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ 𝑧 ∈ ω) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
36	35	iunssd 5009	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
37	5, 36	eqsstrrid 3976	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
38	2, 37	sylan 589	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
39	38	an32s 662	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) ∧ 𝑥 ∈ 𝐴) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
40	39	iunssd 5009	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
41	1, 40	eqsstrid 3975	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  {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:  ttcss  36863  ttcel  36865  ttctrid  36867  dfttc2g  36871  ttcuniun  36875  ttciunun  36876  ttcuni  36878  ttcpwss  36880  ttcsnmin  36883  dfttc3gw  36888  ttcwf  36889  dfttc4  36895  ttcexg  36897