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Theorem rtrclreclem2 15066
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 12541 . . . . 5 0 ∈ ℕ0
2 ssid 4002 . . . . . 6 ( I ↾ 𝑅) ⊆ ( I ↾ 𝑅)
3 rtrclreclem2.1 . . . . . . 7 (𝜑 → Rel 𝑅)
4 rtrclreclem2.2 . . . . . . 7 (𝜑𝑅𝑉)
53, 4relexp0d 15031 . . . . . 6 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
62, 5sseqtrrid 4033 . . . . 5 (𝜑 → ( I ↾ 𝑅) ⊆ (𝑅𝑟0))
7 oveq2 7434 . . . . . . 7 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
87sseq2d 4012 . . . . . 6 (𝑛 = 0 → (( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) ↔ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)))
98rspcev 3608 . . . . 5 ((0 ∈ ℕ0 ∧ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
101, 6, 9sylancr 585 . . . 4 (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
11 ssiun 5056 . . . 4 (∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1210, 11syl 17 . . 3 (𝜑 → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
134elexd 3485 . . . 4 (𝜑𝑅 ∈ V)
14 nn0ex 12532 . . . . 5 0 ∈ V
15 ovex 7459 . . . . 5 (𝑅𝑟𝑛) ∈ V
1614, 15iunex 7984 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
17 oveq1 7433 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
1817iuneq2d 5032 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
19 eqid 2726 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
2018, 19fvmptg 7009 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2113, 16, 20sylancl 584 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2212, 21sseqtrrd 4021 . 2 (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
23 df-rtrclrec 15063 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
24 fveq1 6902 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2524sseq2d 4012 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (( I ↾ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2625imbi2d 339 . . 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 230 1 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wrex 3060  Vcvv 3462  wss 3947   cuni 4915   ciun 5003  cmpt 5238   I cid 5581  cres 5686  Rel wrel 5689  cfv 6556  (class class class)co 7426  0cc0 11160  0cn0 12526  𝑟crelexp 15026  t*reccrtrcl 15062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-mulcl 11222  ax-i2m1 11228
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  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 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-2nd 8006  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-nn 12267  df-n0 12527  df-relexp 15027  df-rtrclrec 15063
This theorem is referenced by:  dfrtrcl2  15069
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