Theorem ttcmin 36684

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 36675	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
2		ssel2 3917	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3		rdgfun 8346	. . . . . . 7 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
4		funiunfv 7194	. . . . . . 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 6832	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅))
7	6	sseq1d 3954	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵))
8		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
9	8	sseq1d 3954	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵))
10		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤))
11	10	sseq1d 3954	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵))
12		vsnex 5370	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ V
13	12	rdg0 8351	. . . . . . . . . . 11 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) = {𝑥}
14		snssi 4752	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
15	13, 14	eqsstrid 3961	. . . . . . . . . 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 7814	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
18		fvex 6845	. . . . . . . . . . . . . 14 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
19	18	uniex 7686	. . . . . . . . . . . . 13 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
20		eqid 2737	. . . . . . . . . . . . . 14 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
21		unieq 4862	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦)
22		unieq 4862	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) → ∪ 𝑧 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
23	20, 21, 22	rdgsucmpt2 8360	. . . . . . . . . . . . 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 4859	. . . . . . . . . . . . 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 5194	. . . . . . . . . . . . 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 3933	. . . . . . . . . . 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 3957	. . . . . . . . . 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 7840	. . . . . . . 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 4994	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
37	5, 36	eqsstrrid 3962	. . . . 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 4994	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
41	1, 40	eqsstrid 3961	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851  ∪ ciun 4934   ↦ cmpt 5167  Tr wtr 5193   “ cima 5625  Oncon0 6315  suc csuc 6317  Fun wfun 6484  ‘cfv 6490  ωcom 7808  reccrdg 8339  TC+ cttc 36674

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 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-ttc 36675

This theorem is referenced by:  ttcss  36686  ttcel  36688  ttctrid  36690  dfttc2g  36694  ttcuniun  36698  ttciunun  36699  ttcuni  36701  ttcpwss  36703  ttcsnmin  36706  dfttc3gw  36711  ttcwf  36712  dfttc4  36718  ttcexg  36720