Theorem ttcid 36790

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 4612	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
2		vsnex 5382	. . . . . . 7 ⊢ {𝑧} ∈ V
3	2	rdg0 8376	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) = {𝑧}
4		rdgfnon 8373	. . . . . . 7 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) Fn On
5		omsson 7835	. . . . . . 7 ⊢ ω ⊆ On
6		peano1 7854	. . . . . . 7 ⊢ ∅ ∈ ω
7		fnfvima 7202	. . . . . . 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 1472	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
9	3, 8	eqeltrri 2849	. . . . 5 ⊢ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
10		elunii 4860	. . . . 5 ⊢ ((𝑧 ∈ {𝑧} ∧ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
11	1, 9, 10	mp2an 700	. . . 4 ⊢ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
12		sneq 4582	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
13		rdgeq2 8367	. . . . . . . 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 6034	. . . . . 6 ⊢ (𝑥 = 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
16	15	unieqd 4868	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
17	16	eliuni 4945	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
18	11, 17	mpan2 699	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
19		df-ttc 36785	. . 3 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
20	18, 19	eleqtrrdi 2863	. 2 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ TC+ 𝐴)
21	20	ssriv 3931	1 ⊢ 𝐴 ⊆ TC+ 𝐴

Syntax hints:   = wceq 1550   ∈ wcel 2132  Vcvv 3444   ⊆ wss 3895  ∅c0 4276  {csn 4572  ∪ cuni 4855  ∪ ciun 4939   ↦ cmpt 5171   “ cima 5639  Oncon0 6331   Fn wfn 6501  ‘cfv 6506  ωcom 7831  reccrdg 8364  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:  ttcexrg  36795  ttcss2  36797  ttcel2  36799  ttctrid  36800  dfttc2g  36804  ttc00  36806  ttcuniun  36808  ttciunun  36809  ttcuni  36811  ttcpwss  36813  ttcsnidg  36815  dfttc3gw  36821  ttcwf  36822  dfttc4  36828