Theorem rtrclreclem2 15135

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 12546	. . . . 5 ⊢ 0 ∈ ℕ0
2		ssid 3953	. . . . . 6 ⊢ ( I ↾ ∪ ∪ 𝑅) ⊆ ( I ↾ ∪ ∪ 𝑅)
3		rtrclreclem2.1	. . . . . . 7 ⊢ (𝜑 → Rel 𝑅)
4		rtrclreclem2.2	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5	3, 4	relexp0d 15100	. . . . . 6 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
6	2, 5	sseqtrrid 3974	. . . . 5 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0))
7		oveq2 7422	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
8	7	sseq2d 3963	. . . . . 6 ⊢ (𝑛 = 0 → (( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)))
9	8	rspcev 3576	. . . . 5 ⊢ ((0 ∈ ℕ0 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛))
10	1, 6, 9	sylancr 599	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛))
11		ssiun 5005	. . . 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 3473	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
14		nn0ex 12537	. . . . 5 ⊢ ℕ0 ∈ V
15		ovex 7447	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
16	14, 15	iunex 7966	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
17		oveq1 7421	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
18	17	iuneq2d 4981	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
19		eqid 2760	. . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
20	18, 19	fvmptg 6985	. . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
21	13, 16, 20	sylancl 598	. . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
22	12, 21	sseqtrrd 3968	. 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
23		df-rtrclrec 15132	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
24		fveq1 6878	. . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
25	24	sseq2d 3963	. . . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186   I cid 5549   ↾ cres 5657  Rel wrel 5660  ‘cfv 6533  (class class class)co 7414  0cc0 11127  ℕ₀cn0 12531  ↑_𝑟crelexp 15095  t*reccrtrcl 15131

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 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-mulcl 11189  ax-i2m1 11195

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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-nn 12261  df-n0 12532  df-relexp 15096  df-rtrclrec 15132

This theorem is used by:  dfrtrcl2  15138