Theorem ttcid 37202

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 4623	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
2		vsnex 5392	. . . . . . 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 7864	. . . . . . 7 ⊢ ω ⊆ On
6		peano1 7883	. . . . . . 7 ⊢ ∅ ∈ ω
7		fnfvima 7227	. . . . . . 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 1490	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
9	3, 8	eqeltrri 2857	. . . . 5 ⊢ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
10		elunii 4871	. . . . 5 ⊢ ((𝑧 ∈ {𝑧} ∧ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
11	1, 9, 10	mp2an 705	. . . 4 ⊢ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
12		sneq 4593	. . . . . . . 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 6049	. . . . . 6 ⊢ (𝑥 = 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
16	15	unieqd 4879	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
17	16	eliuni 4956	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)) → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
18	11, 17	mpan2 704	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
19		df-ttc 37197	. . 3 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
20	18, 19	eleqtrrdi 2871	. 2 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ TC+ 𝐴)
21	20	ssriv 3934	1 ⊢ 𝐴 ⊆ TC+ 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185   “ cima 5650  Oncon0 6351   Fn wfn 6522  ‘cfv 6527  ωcom 7860  reccrdg 8395  TC+ cttc 37196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 37197

This theorem is used by:  ttcexrg  37207  ttcss2  37209  ttcel2  37211  ttctrid  37212  dfttc2g  37216  ttc00  37218  ttcuniun  37220  ttciunun  37221  ttcuni  37223  ttcpwss  37225  ttcsnidg  37227  dfttc3gw  37233  ttcwf  37234  dfttc4  37240