Theorem ttciunun 36809

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 4122	. . 3 ⊢ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
2		dftr3 5202	. . . 4 ⊢ (Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ ∀𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
3		elun 4097	. . . . . 6 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴))
4		ttctr 36791	. . . . . . . . 9 ⊢ Tr TC+ 𝑥
5	4	rgenw 3070	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 Tr TC+ 𝑥
6		triun 5212	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr TC+ 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
7		trss 5207	. . . . . . . 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 36790	. . . . . . . 8 ⊢ 𝑦 ⊆ TC+ 𝑦
10		ttceq 36786	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → TC+ 𝑥 = TC+ 𝑦)
11	10	ssiun2s 4996	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → TC+ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
12	9, 11	sstrid 3938	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
13	8, 12	jaoi 866	. . . . . 6 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
14	3, 13	sylbi 219	. . . . 5 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
15		ssun3 4123	. . . . 5 ⊢ (𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 → 𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
16	14, 15	syl 17	. . . 4 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) → 𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
17	2, 16	mprgbir 3073	. . 3 ⊢ Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
18		ttcmin 36794	. . 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 700	. 2 ⊢ TC+ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
20		iunss 4992	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴 ↔ ∀𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴)
21		ttcel2 36799	. . . 4 ⊢ (𝑥 ∈ 𝐴 → TC+ 𝑥 ⊆ TC+ 𝐴)
22	20, 21	mprgbir 3073	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴
23		ttcid 36790	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
24	22, 23	unssi 4134	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ⊆ TC+ 𝐴
25	19, 24	eqssi 3943	1 ⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

Syntax hints:   → wi 4   ∨ wo 856   = wceq 1550   ∈ wcel 2132  ∀wral 3066   ∪ cun 3893   ⊆ wss 3895  ∪ ciun 4939  Tr wtr 5197  TC+ cttc 36784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-ttc 36785

This theorem is referenced by:  ttcun  36810  ttciun  36812  ttcsng  36817