Theorem ttcmin 37035

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 37026	. 2 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
2		ssel2 3931	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3		rdgfun 8401	. . . . . . 7 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
4		funiunfv 7246	. . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅))
7	6	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵))
8		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
9	8	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵))
10		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑤 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤))
11	10	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑤 → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵 ↔ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) ⊆ 𝐵))
12		vsnex 5405	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ V
13	12	rdg0 8406	. . . . . . . . . . 11 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) = {𝑥}
14		snssi 4750	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
15	13, 14	eqsstrid 3974	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
16	15	adantr 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘∅) ⊆ 𝐵)
17		nnon 7866	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
18		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
19	18	uniex 7741	. . . . . . . . . . . . 13 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ∈ V
20		eqid 2762	. . . . . . . . . . . . . 14 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
21		unieq 4882	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦)
22		unieq 4882	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) → ∪ 𝑧 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
23	20, 21, 22	rdgsucmpt2 8415	. . . . . . . . . . . . 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 597	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑤) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
25	24	3ad2ant1 1150	. . . . . . . . . . 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 4879	. . . . . . . . . . . . 13 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
27	26	3ad2ant3 1152	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ ∪ 𝐵)
28		simp2r 1218	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ (𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ⊆ 𝐵) → Tr 𝐵)
29		df-tr 5218	. . . . . . . . . . . . 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 3946	. . . . . . . . . . 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 3970	. . . . . . . . . 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 1136	. . . . . . . . 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 7893	. . . . . . . 8 ⊢ (𝑧 ∈ ω → ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵))
35	34	impcom 412	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ Tr 𝐵) ∧ 𝑧 ∈ ω) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
36	35	iunssd 5014	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ 𝑧 ∈ ω (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ⊆ 𝐵)
37	5, 36	eqsstrrid 3975	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
38	2, 37	sylan 591	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ Tr 𝐵) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
39	38	an32s 664	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) ∧ 𝑥 ∈ 𝐴) → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
40	39	iunssd 5014	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ⊆ 𝐵)
41	1, 40	eqsstrid 3974	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191  Tr wtr 5217   “ cima 5663  Oncon0 6360  suc csuc 6362  Fun wfun 6530  ‘cfv 6536  ωcom 7860  reccrdg 8394  TC+ cttc 37025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-ttc 37026

This theorem is used by:  ttcss  37037  ttcel  37039  ttctrid  37041  dfttc2g  37045  ttcuniun  37049  ttciunun  37050  ttcuni  37052  ttcpwss  37054  ttcsnmin  37057  dfttc3gw  37062  ttcwf  37063  dfttc4  37069  ttcexg  37071