Theorem rtrclreclem2 14698

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 12178	. . . . 5 ⊢ 0 ∈ ℕ0
2		ssid 3939	. . . . . 6 ⊢ ( I ↾ ∪ ∪ 𝑅) ⊆ ( I ↾ ∪ ∪ 𝑅)
3		rtrclreclem2.1	. . . . . . 7 ⊢ (𝜑 → Rel 𝑅)
4		rtrclreclem2.2	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5	3, 4	relexp0d 14663	. . . . . 6 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
6	2, 5	sseqtrrid 3970	. . . . 5 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0))
7		oveq2 7263	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
8	7	sseq2d 3949	. . . . . 6 ⊢ (𝑛 = 0 → (( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)))
9	8	rspcev 3552	. . . . 5 ⊢ ((0 ∈ ℕ0 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛))
10	1, 6, 9	sylancr 586	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛))
11		ssiun 4972	. . . 4 ⊢ (∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
13	4	elexd 3442	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
14		nn0ex 12169	. . . . 5 ⊢ ℕ0 ∈ V
15		ovex 7288	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
16	14, 15	iunex 7784	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
17		oveq1 7262	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
18	17	iuneq2d 4950	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
19		eqid 2738	. . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
20	18, 19	fvmptg 6855	. . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
21	13, 16, 20	sylancl 585	. . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
22	12, 21	sseqtrrd 3958	. 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
23		df-rtrclrec 14695	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
24		fveq1 6755	. . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
25	24	sseq2d 3949	. . . 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 340	. . 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 230	1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∪ cuni 4836  ∪ ciun 4921   ↦ cmpt 5153   I cid 5479   ↾ cres 5582  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  0cc0 10802  ℕ₀cn0 12163  ↑_𝑟crelexp 14658  t*reccrtrcl 14694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-mulcl 10864  ax-i2m1 10870

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-nn 11904  df-n0 12164  df-relexp 14659  df-rtrclrec 14695

This theorem is referenced by:  dfrtrcl2  14701