Theorem ttciunun 36699

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 4120	. . 3 ⊢ 𝐴 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
2		dftr3 5198	. . . 4 ⊢ (Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ ∀𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)𝑦 ⊆ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴))
3		elun 4094	. . . . . 6 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∨ 𝑦 ∈ 𝐴))
4		ttctr 36681	. . . . . . . . 9 ⊢ Tr TC+ 𝑥
5	4	rgenw 3056	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 Tr TC+ 𝑥
6		triun 5207	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr TC+ 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
7		trss 5203	. . . . . . . 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 36680	. . . . . . . 8 ⊢ 𝑦 ⊆ TC+ 𝑦
10		ttceq 36676	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → TC+ 𝑥 = TC+ 𝑦)
11	10	ssiun2s 4992	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → TC+ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥)
12	9, 11	sstrid 3934	. . . . . . 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 4121	. . . . 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 3059	. . 3 ⊢ Tr (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
18		ttcmin 36684	. . 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 4988	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴 ↔ ∀𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴)
21		ttcel2 36689	. . . 4 ⊢ (𝑥 ∈ 𝐴 → TC+ 𝑥 ⊆ TC+ 𝐴)
22	20, 21	mprgbir 3059	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ⊆ TC+ 𝐴
23		ttcid 36680	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
24	22, 23	unssi 4132	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ⊆ TC+ 𝐴
25	19, 24	eqssi 3939	1 ⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3888   ⊆ wss 3890  ∪ ciun 4934  Tr wtr 5193  TC+ cttc 36674

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-ttc 36675

This theorem is referenced by:  ttcun  36700  ttciun  36702  ttcsng  36707