Theorem ttcmin 36666

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 36657	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
2		ssel2 3912	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3		rdgfun 8344	. . . . . . 7 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
4		funiunfv 7192	. . . . . . 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 6829	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅))
7	6	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵))
8		fveq2 6829	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
9	8	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵))
10		fveq2 6829	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤))
11	10	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵))
12		vsnex 5366	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ V
13	12	rdg0 8349	. . . . . . . . . . 11 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) = {𝑥}
14		snssi 4719	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
15	13, 14	eqsstrid 3955	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
17		nnon 7812	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
18		fvex 6842	. . . . . . . . . . . . . 14 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
19	18	uniex 7684	. . . . . . . . . . . . 13 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
20		eqid 2735	. . . . . . . . . . . . . 14 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
21		unieq 4851	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦)
22		unieq 4851	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) → ∪ 𝑧 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
23	20, 21, 22	rdgsucmpt2 8358	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
25	24	3ad2ant1 1134	. . . . . . . . . . 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 4848	. . . . . . . . . . . . 13 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
27	26	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
28		simp2r 1202	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → Tr 𝐵)
29		df-tr 5182	. . . . . . . . . . . . 13 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵)
30	28, 29	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ 𝐵 ⊆ 𝐵)
31	27, 30	sstrd 3927	. . . . . . . . . . 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 3951	. . . . . . . . . 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 1120	. . . . . . . . 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 7838	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵))
35	34	impcom 407	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ 𝑧 ∈ ω) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
36	35	iunssd 4982	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
37	5, 36	eqsstrrid 3956	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
38	2, 37	sylan 581	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
39	38	an32s 653	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) ∧ 𝑥 ∈ 𝐴) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
40	39	iunssd 4982	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
41	1, 40	eqsstrid 3955	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3427   ⊆ wss 3885  ∅c0 4263  {csn 4557  ∪ cuni 4840  ∪ ciun 4923   ↦ cmpt 5155  Tr wtr 5181   “ cima 5623  Oncon0 6312  suc csuc 6314  Fun wfun 6481  ‘cfv 6487  ωcom 7806  reccrdg 8337  TC+ cttc 36656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  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 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-ttc 36657

This theorem is referenced by:  ttcss  36668  ttcel  36670  ttctrid  36672  dfttc2g  36676  ttcuniun  36680  ttciunun  36681  ttcuni  36683  ttcpwss  36685  ttcsnmin  36688  dfttc3gw  36693  ttcwf  36694  dfttc4  36700  ttcexg  36702