Theorem ttcid 36735

Description: The transitive closure contains its argument as a subclass. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcid	⊢ 𝐴 ⊆ TC+ 𝐴

Proof of Theorem ttcid

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vsnid 4598	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
2		vsnex 5367	. . . . . . 7 ⊢ {𝑧} ∈ V
3	2	rdg0 8354	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) = {𝑧}
4		rdgfnon 8351	. . . . . . 7 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) Fn On
5		omsson 7814	. . . . . . 7 ⊢ ω ⊆ On
6		peano1 7833	. . . . . . 7 ⊢ ∅ ∈ ω
7		fnfvima 7181	. . . . . . 7 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) Fn On ∧ ω ⊆ On ∧ ∅ ∈ ω) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
8	4, 5, 6, 7	mp3an 1470	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
9	3, 8	eqeltrri 2838	. . . . 5 ⊢ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
10		elunii 4846	. . . . 5 ⊢ ((𝑧 ∈ {𝑧} ∧ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
11	1, 9, 10	mp2an 699	. . . 4 ⊢ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
12		sneq 4568	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
13		rdgeq2 8345	. . . . . . . 8 ⊢ ({𝑥} = {𝑧} → rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}))
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝑧 → rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}))
15	14	imaeq1d 6018	. . . . . 6 ⊢ (𝑥 = 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
16	15	unieqd 4854	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
17	16	eliuni 4930	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
18	11, 17	mpan2 698	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
19		df-ttc 36730	. . 3 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
20	18, 19	eleqtrrdi 2852	. 2 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ TC+ 𝐴)
21	20	ssriv 3921	1 ⊢ 𝐴 ⊆ TC+ 𝐴

Syntax hints:   = wceq 1548   ∈ wcel 2121  Vcvv 3433   ⊆ wss 3885  ∅c0 4264  {csn 4558  ∪ cuni 4841  ∪ ciun 4924   ↦ cmpt 5156   “ cima 5624  Oncon0 6314   Fn wfn 6484  ‘cfv 6489  ωcom 7810  reccrdg 8342  TC+ cttc 36729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-ttc 36730

This theorem is referenced by:  ttcexrg  36740  ttcss2  36742  ttcel2  36744  ttctrid  36745  dfttc2g  36749  ttc00  36751  ttcuniun  36753  ttciunun  36754  ttcuni  36756  ttcpwss  36758  ttcsnidg  36760  dfttc3gw  36766  ttcwf  36767  dfttc4  36773