Theorem ttcid 36947

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 4628	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
2		vsnex 5406	. . . . . . 7 ⊢ {𝑧} ∈ V
3	2	rdg0 8407	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) = {𝑧}
4		rdgfnon 8404	. . . . . . 7 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) Fn On
5		omsson 7865	. . . . . . 7 ⊢ ω ⊆ On
6		peano1 7884	. . . . . . 7 ⊢ ∅ ∈ ω
7		fnfvima 7231	. . . . . . 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 1488	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
9	3, 8	eqeltrri 2858	. . . . 5 ⊢ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
10		elunii 4876	. . . . 5 ⊢ ((𝑧 ∈ {𝑧} ∧ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
11	1, 9, 10	mp2an 704	. . . 4 ⊢ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
12		sneq 4598	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
13		rdgeq2 8398	. . . . . . . 8 ⊢ ({𝑥} = {𝑧} → rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}))
14	12, 13	syl 18	. . . . . . 7 ⊢ (𝑥 = 𝑧 → rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}))
15	14	imaeq1d 6061	. . . . . 6 ⊢ (𝑥 = 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
16	15	unieqd 4884	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
17	16	eliuni 4961	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
18	11, 17	mpan2 703	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
19		df-ttc 36942	. . 3 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
20	18, 19	eleqtrrdi 2872	. 2 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ TC+ 𝐴)
21	20	ssriv 3940	1 ⊢ 𝐴 ⊆ TC+ 𝐴

Syntax hints:   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191   “ cima 5664  Oncon0 6360   Fn wfn 6531  ‘cfv 6536  ωcom 7861  reccrdg 8395  TC+ cttc 36941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 36942

This theorem is referenced by:  ttcexrg  36952  ttcss2  36954  ttcel2  36956  ttctrid  36957  dfttc2g  36961  ttc00  36963  ttcuniun  36965  ttciunun  36966  ttcuni  36968  ttcpwss  36970  ttcsnidg  36972  dfttc3gw  36978  ttcwf  36979  dfttc4  36985