Theorem ttciunun 37221

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 4124	. . 3 ⊢ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
2		dftr3 5216	. . . 4 ⊢ (Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ ∀𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
3		elun 4099	. . . . . 6 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴))
4		ttctr 37203	. . . . . . . . 9 ⊢ Tr TC+ 𝑥
5	4	rgenw 3080	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 Tr TC+ 𝑥
6		triun 5226	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr TC+ 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
7		trss 5221	. . . . . . . 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 37202	. . . . . . . 8 ⊢ 𝑦 ⊆ TC+ 𝑦
10		ttceq 37198	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → TC+ 𝑥 = TC+ 𝑦)
11	10	ssiun2s 5006	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → TC+ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
12	9, 11	sstrid 3941	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
13	8, 12	jaoi 871	. . . . . 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 4125	. . . . 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 3083	. . 3 ⊢ Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
18		ttcmin 37206	. . 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 705	. 2 ⊢ TC+ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
20		iunss 5002	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴 ↔ ∀𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴)
21		ttcel2 37211	. . . 4 ⊢ (𝑥 ∈ 𝐴 → TC+ 𝑥 ⊆ TC+ 𝐴)
22	20, 21	mprgbir 3083	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴
23		ttcid 37202	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
24	22, 23	unssi 4136	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ⊆ TC+ 𝐴
25	19, 24	eqssi 3946	1 ⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∪ cun 3896   ⊆ wss 3898  ∪ ciun 4950  Tr wtr 5211  TC+ cttc 37196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 37197

This theorem is used by:  ttcun  37222  ttciun  37224  ttcsng  37229