Theorem ttciunun 37050

Description: Relationship between TC+ 𝐴 and ∪ 𝑥 ∈ 𝐴TC+ 𝑥: we can decompose TC+ 𝐴 into the elements of ∪ 𝑥 ∈ 𝐴TC+ 𝑥 plus the elements of 𝐴 itself. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttciunun	⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem ttciunun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun2 4131	. . 3 ⊢ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
2		dftr3 5222	. . . 4 ⊢ (Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ ∀𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
3		elun 4106	. . . . . 6 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴))
4		ttctr 37032	. . . . . . . . 9 ⊢ Tr TC+ 𝑥
5	4	rgenw 3082	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 Tr TC+ 𝑥
6		triun 5232	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr TC+ 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
7		trss 5227	. . . . . . . 8 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥))
8	5, 6, 7	mp2b 10	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
9		ttcid 37031	. . . . . . . 8 ⊢ 𝑦 ⊆ TC+ 𝑦
10		ttceq 37027	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → TC+ 𝑥 = TC+ 𝑦)
11	10	ssiun2s 5012	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → TC+ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
12	9, 11	sstrid 3947	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
13	8, 12	jaoi 870	. . . . . 6 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
14	3, 13	sylbi 220	. . . . 5 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
15		ssun3 4132	. . . . 5 ⊢ (𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 → 𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
16	14, 15	syl 18	. . . 4 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) → 𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
17	2, 16	mprgbir 3085	. . 3 ⊢ Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
18		ttcmin 37035	. . 3 ⊢ ((𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ∧ Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)) → TC+ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
19	1, 17, 18	mp2an 704	. 2 ⊢ TC+ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
20		iunss 5008	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴 ↔ ∀𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴)
21		ttcel2 37040	. . . 4 ⊢ (𝑥 ∈ 𝐴 → TC+ 𝑥 ⊆ TC+ 𝐴)
22	20, 21	mprgbir 3085	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴
23		ttcid 37031	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
24	22, 23	unssi 4143	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ⊆ TC+ 𝐴
25	19, 24	eqssi 3952	1 ⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∨ wo 860   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ∪ cun 3902   ⊆ wss 3904  ∪ ciun 4955  Tr wtr 5217  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:  ttcun  37051  ttciun  37053  ttcsng  37058