Theorem ttciunun 36693

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 4119	. . 3 ⊢ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
2		dftr3 5197	. . . 4 ⊢ (Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ ∀𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
3		elun 4093	. . . . . 6 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴))
4		ttctr 36675	. . . . . . . . 9 ⊢ Tr TC+ 𝑥
5	4	rgenw 3055	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 Tr TC+ 𝑥
6		triun 5207	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr TC+ 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
7		trss 5202	. . . . . . . 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 36674	. . . . . . . 8 ⊢ 𝑦 ⊆ TC+ 𝑦
10		ttceq 36670	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → TC+ 𝑥 = TC+ 𝑦)
11	10	ssiun2s 4991	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → TC+ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
12	9, 11	sstrid 3933	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
13	8, 12	jaoi 858	. . . . . 6 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
14	3, 13	sylbi 217	. . . . 5 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
15		ssun3 4120	. . . . 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 3058	. . 3 ⊢ Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
18		ttcmin 36678	. . 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 693	. 2 ⊢ TC+ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
20		iunss 4987	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴 ↔ ∀𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴)
21		ttcel2 36683	. . . 4 ⊢ (𝑥 ∈ 𝐴 → TC+ 𝑥 ⊆ TC+ 𝐴)
22	20, 21	mprgbir 3058	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴
23		ttcid 36674	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
24	22, 23	unssi 4131	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ⊆ TC+ 𝐴
25	19, 24	eqssi 3938	1 ⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ∪ cun 3887   ⊆ wss 3889  ∪ ciun 4933  Tr wtr 5192  TC+ cttc 36668

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-ttc 36669

This theorem is referenced by:  ttcun  36694  ttciun  36696  ttcsng  36701