Theorem ttcid 37098

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 4627	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
2		vsnex 5404	. . . . . . 7 ⊢ {𝑧} ∈ V
3	2	rdg0 8413	. . . . . 6 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧})‘∅) = {𝑧}
4		rdgfnon 8410	. . . . . . 7 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) Fn On
5		omsson 7869	. . . . . . 7 ⊢ ω ⊆ On
6		peano1 7888	. . . . . . 7 ⊢ ∅ ∈ ω
7		fnfvima 7235	. . . . . . 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 2859	. . . . 5 ⊢ {𝑧} ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω)
10		elunii 4875	. . . . 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 4597	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
13		rdgeq2 8404	. . . . . . . 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 6059	. . . . . 6 ⊢ (𝑥 = 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
16	15	unieqd 4883	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑧}) “ ω))
17	16	eliuni 4960	. . . 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 37093	. . 3 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
20	18, 19	eleqtrrdi 2873	. 2 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ TC+ 𝐴)
21	20	ssriv 3938	1 ⊢ 𝐴 ⊆ TC+ 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  {csn 4587  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5190   “ cima 5662  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  ωcom 7865  reccrdg 8401  TC+ cttc 37092

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-ttc 37093

This theorem is used by:  ttcexrg  37103  ttcss2  37105  ttcel2  37107  ttctrid  37108  dfttc2g  37112  ttc00  37114  ttcuniun  37116  ttciunun  37117  ttcuni  37119  ttcpwss  37121  ttcsnidg  37123  dfttc3gw  37129  ttcwf  37130  dfttc4  37136