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Theorem rtrclreclem2 14410
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 11900 . . . . 5 0 ∈ ℕ0
2 ssid 3937 . . . . . 6 ( I ↾ 𝑅) ⊆ ( I ↾ 𝑅)
3 rtrclreclem2.1 . . . . . . 7 (𝜑 → Rel 𝑅)
4 rtrclreclem2.2 . . . . . . 7 (𝜑𝑅𝑉)
53, 4relexp0d 14375 . . . . . 6 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
62, 5sseqtrrid 3968 . . . . 5 (𝜑 → ( I ↾ 𝑅) ⊆ (𝑅𝑟0))
7 oveq2 7143 . . . . . . 7 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
87sseq2d 3947 . . . . . 6 (𝑛 = 0 → (( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) ↔ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)))
98rspcev 3571 . . . . 5 ((0 ∈ ℕ0 ∧ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
101, 6, 9sylancr 590 . . . 4 (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
11 ssiun 4933 . . . 4 (∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1210, 11syl 17 . . 3 (𝜑 → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
134elexd 3461 . . . 4 (𝜑𝑅 ∈ V)
14 nn0ex 11891 . . . . 5 0 ∈ V
15 ovex 7168 . . . . 5 (𝑅𝑟𝑛) ∈ V
1614, 15iunex 7651 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
17 oveq1 7142 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
1817iuneq2d 4910 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
19 eqid 2798 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
2018, 19fvmptg 6743 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2113, 16, 20sylancl 589 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2212, 21sseqtrrd 3956 . 2 (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
23 df-rtrclrec 14407 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
24 fveq1 6644 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2524sseq2d 3947 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (( I ↾ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2625imbi2d 344 . . 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 234 1 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2111  wrex 3107  Vcvv 3441  wss 3881   cuni 4800   ciun 4881  cmpt 5110   I cid 5424  cres 5521  Rel wrel 5524  cfv 6324  (class class class)co 7135  0cc0 10526  0cn0 11885  𝑟crelexp 14370  t*reccrtrcl 14406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-mulcl 10588  ax-i2m1 10594
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-nn 11626  df-n0 11886  df-relexp 14371  df-rtrclrec 14407
This theorem is referenced by:  dfrtrcl2  14413
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