Theorem ttcmin 37326

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 37317	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
2		ssel2 3903	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3		rdgfun 8424	. . . . . . 7 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
4		funiunfv 7252	. . . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅))
7	6	sseq1d 3939	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵))
8		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
9	8	sseq1d 3939	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵))
10		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤))
11	10	sseq1d 3939	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵))
12		vsnex 5392	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ V
13	12	rdg0 8429	. . . . . . . . . . 11 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) = {𝑥}
14		snssi 4745	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
15	13, 14	eqsstrid 3946	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
17		nnon 7883	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
18		fvex 6898	. . . . . . . . . . . . . 14 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
19	18	uniex 7758	. . . . . . . . . . . . 13 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
20		eqid 2760	. . . . . . . . . . . . . 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 8438	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
25	24	3ad2ant1 1151	. . . . . . . . . . 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 1153	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
28		simp2r 1219	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → Tr 𝐵)
29		df-tr 5212	. . . . . . . . . . . . 13 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵)
30	28, 29	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ 𝐵 ⊆ 𝐵)
31	27, 30	sstrd 3918	. . . . . . . . . . 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 3942	. . . . . . . . . 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 1137	. . . . . . . . 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 7910	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵))
35	34	impcom 413	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ 𝑧 ∈ ω) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
36	35	iunssd 5008	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
37	5, 36	eqsstrrid 3947	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
38	2, 37	sylan 592	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
39	38	an32s 665	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) ∧ 𝑥 ∈ 𝐴) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
40	39	iunssd 5008	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
41	1, 40	eqsstrid 3946	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185  Tr wtr 5211   “ cima 5653  Oncon0 6362  suc csuc 6364  Fun wfun 6532  ‘cfv 6538  ωcom 7877  reccrdg 8417  TC+ cttc 37316

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-ttc 37317

This theorem is used by:  ttcss  37328  ttcel  37330  ttctrid  37332  dfttc2g  37336  ttcuniun  37340  ttciunun  37341  ttcuni  37343  ttcpwss  37345  ttcsnmin  37348  dfttc3gw  37353  ttcwf  37354  dfttc4  37360  ttcexg  37362