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Theorem rtrclreclem2 15099
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 (𝜑𝑅𝑉)
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.
StepHypRef Expression
1 0nn0 12543 . . . . 5 0 ∈ ℕ0
2 ssid 4005 . . . . . 6 ( I ↾ 𝑅) ⊆ ( I ↾ 𝑅)
3 rtrclreclem2.1 . . . . . . 7 (𝜑 → Rel 𝑅)
4 rtrclreclem2.2 . . . . . . 7 (𝜑𝑅𝑉)
53, 4relexp0d 15064 . . . . . 6 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
62, 5sseqtrrid 4026 . . . . 5 (𝜑 → ( I ↾ 𝑅) ⊆ (𝑅𝑟0))
7 oveq2 7440 . . . . . . 7 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
87sseq2d 4015 . . . . . 6 (𝑛 = 0 → (( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) ↔ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)))
98rspcev 3621 . . . . 5 ((0 ∈ ℕ0 ∧ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
101, 6, 9sylancr 587 . . . 4 (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
11 ssiun 5045 . . . 4 (∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1210, 11syl 17 . . 3 (𝜑 → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
134elexd 3503 . . . 4 (𝜑𝑅 ∈ V)
14 nn0ex 12534 . . . . 5 0 ∈ V
15 ovex 7465 . . . . 5 (𝑅𝑟𝑛) ∈ V
1614, 15iunex 7994 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
17 oveq1 7439 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
1817iuneq2d 5021 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
19 eqid 2736 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
2018, 19fvmptg 7013 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2113, 16, 20sylancl 586 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2212, 21sseqtrrd 4020 . 2 (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
23 df-rtrclrec 15096 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
24 fveq1 6904 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2524sseq2d 4015 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (( I ↾ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2625imbi2d 340 . . 3 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))))
2723, 26ax-mp 5 . 2 ((𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2822, 27mpbir 231 1 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  wrex 3069  Vcvv 3479  wss 3950   cuni 4906   ciun 4990  cmpt 5224   I cid 5576  cres 5686  Rel wrel 5689  cfv 6560  (class class class)co 7432  0cc0 11156  0cn0 12528  𝑟crelexp 15059  t*reccrtrcl 15095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-mulcl 11218  ax-i2m1 11224
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-nn 12268  df-n0 12529  df-relexp 15060  df-rtrclrec 15096
This theorem is referenced by:  dfrtrcl2  15102
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