Theorem rtrclreclem2 15092

Description: The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)

Hypotheses

Ref	Expression
rtrclreclem2.1	⊢ (𝜑 → Rel 𝑅)
rtrclreclem2.2	⊢ (𝜑 → 𝑅 ∈ 𝑉)

Assertion

Ref	Expression
rtrclreclem2	⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))

Proof of Theorem rtrclreclem2

Dummy variables 𝑟 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nn0 12514	. . . . 5 ⊢ 0 ∈ ℕ0
2		ssid 3959	. . . . . 6 ⊢ ( I ↾ ∪ ∪ 𝑅) ⊆ ( I ↾ ∪ ∪ 𝑅)
3		rtrclreclem2.1	. . . . . . 7 ⊢ (𝜑 → Rel 𝑅)
4		rtrclreclem2.2	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5	3, 4	relexp0d 15057	. . . . . 6 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
6	2, 5	sseqtrrid 3980	. . . . 5 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0))
7		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
8	7	sseq2d 3969	. . . . . 6 ⊢ (𝑛 = 0 → (( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)))
9	8	rspcev 3581	. . . . 5 ⊢ ((0 ∈ ℕ0 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛))
10	1, 6, 9	sylancr 598	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛))
11		ssiun 5011	. . . 4 ⊢ (∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
12	10, 11	syl 18	. . 3 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
13	4	elexd 3478	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
14		nn0ex 12505	. . . . 5 ⊢ ℕ0 ∈ V
15		ovex 7443	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
16	14, 15	iunex 7961	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
17		oveq1 7417	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
18	17	iuneq2d 4987	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
19		eqid 2763	. . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
20	18, 19	fvmptg 6987	. . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
21	13, 16, 20	sylancl 597	. . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
22	12, 21	sseqtrrd 3974	. 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
23		df-rtrclrec 15089	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
24		fveq1 6880	. . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
25	24	sseq2d 3969	. . . 4 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)))
26	25	imbi2d 343	. . 3 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))))
27	23, 26	ax-mp 5	. 2 ⊢ ((𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)))
28	22, 27	mpbir 234	1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4872  ∪ ciun 4956   ↦ cmpt 5192   I cid 5555   ↾ cres 5663  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410  0cc0 11095  ℕ₀cn0 12499  ↑_𝑟crelexp 15052  t*reccrtrcl 15088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-mulcl 11157  ax-i2m1 11163

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  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-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-nn 12229  df-n0 12500  df-relexp 15053  df-rtrclrec 15089

This theorem is referenced by:  dfrtrcl2  15095