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Theorem rtrclreclem1 13989
Description: The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
Hypotheses
Ref Expression
rtrclreclem.1 (𝜑 → Rel 𝑅)
rtrclreclem.2 (𝜑𝑅 ∈ V)
Assertion
Ref Expression
rtrclreclem1 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))

Proof of Theorem rtrclreclem1
Dummy variables 𝑟 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nn0 11491 . . . . 5 0 ∈ ℕ0
2 ssid 3757 . . . . . 6 ( I ↾ 𝑅) ⊆ ( I ↾ 𝑅)
3 rtrclreclem.1 . . . . . . 7 (𝜑 → Rel 𝑅)
4 rtrclreclem.2 . . . . . . 7 (𝜑𝑅 ∈ V)
53, 4relexp0d 13955 . . . . . 6 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
62, 5syl5sseqr 3787 . . . . 5 (𝜑 → ( I ↾ 𝑅) ⊆ (𝑅𝑟0))
7 oveq2 6813 . . . . . . 7 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
87sseq2d 3766 . . . . . 6 (𝑛 = 0 → (( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) ↔ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)))
98rspcev 3441 . . . . 5 ((0 ∈ ℕ0 ∧ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
101, 6, 9sylancr 698 . . . 4 (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
11 ssiun 4706 . . . 4 (∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1210, 11syl 17 . . 3 (𝜑 → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
13 nn0ex 11482 . . . . 5 0 ∈ V
14 ovex 6833 . . . . 5 (𝑅𝑟𝑛) ∈ V
1513, 14iunex 7304 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
16 oveq1 6812 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
1716iuneq2d 4691 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
18 eqid 2752 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
1917, 18fvmptg 6434 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
204, 15, 19sylancl 697 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2112, 20sseqtr4d 3775 . 2 (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
22 df-rtrclrec 13987 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
23 fveq1 6343 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2423sseq2d 3766 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (( I ↾ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2524imbi2d 329 . . 3 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))))
2622, 25ax-mp 5 . 2 ((𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2721, 26mpbir 221 1 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1624  wcel 2131  wrex 3043  Vcvv 3332  wss 3707   cuni 4580   ciun 4664  cmpt 4873   I cid 5165  cres 5260  Rel wrel 5263  cfv 6041  (class class class)co 6805  0cc0 10120  0cn0 11476  𝑟crelexp 13951  t*reccrtrcl 13986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-i2m1 10188  ax-1ne0 10189  ax-rrecex 10192  ax-cnre 10193
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-nn 11205  df-n0 11477  df-relexp 13952  df-rtrclrec 13987
This theorem is referenced by:  dfrtrcl2  13993
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