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Theorem rtrclreclem1 14278
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 11724 . . . . 5 0 ∈ ℕ0
2 ssid 3880 . . . . . 6 ( I ↾ 𝑅) ⊆ ( I ↾ 𝑅)
3 rtrclreclem.1 . . . . . . 7 (𝜑 → Rel 𝑅)
4 rtrclreclem.2 . . . . . . 7 (𝜑𝑅 ∈ V)
53, 4relexp0d 14244 . . . . . 6 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
62, 5syl5sseqr 3911 . . . . 5 (𝜑 → ( I ↾ 𝑅) ⊆ (𝑅𝑟0))
7 oveq2 6984 . . . . . . 7 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
87sseq2d 3890 . . . . . 6 (𝑛 = 0 → (( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) ↔ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)))
98rspcev 3536 . . . . 5 ((0 ∈ ℕ0 ∧ ( I ↾ 𝑅) ⊆ (𝑅𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
101, 6, 9sylancr 578 . . . 4 (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛))
11 ssiun 4836 . . . 4 (∃𝑛 ∈ ℕ0 ( I ↾ 𝑅) ⊆ (𝑅𝑟𝑛) → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1210, 11syl 17 . . 3 (𝜑 → ( I ↾ 𝑅) ⊆ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
13 nn0ex 11714 . . . . 5 0 ∈ V
14 ovex 7008 . . . . 5 (𝑅𝑟𝑛) ∈ V
1513, 14iunex 7481 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
16 oveq1 6983 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
1716iuneq2d 4820 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
18 eqid 2779 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
1917, 18fvmptg 6593 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
204, 15, 19sylancl 577 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
2112, 20sseqtr4d 3899 . 2 (𝜑 → ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
22 df-rtrclrec 14276 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
23 fveq1 6498 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2423sseq2d 3890 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (( I ↾ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)))
2524imbi2d 333 . . 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 223 1 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wcel 2050  wrex 3090  Vcvv 3416  wss 3830   cuni 4712   ciun 4792  cmpt 5008   I cid 5311  cres 5409  Rel wrel 5412  cfv 6188  (class class class)co 6976  0cc0 10335  0cn0 11707  𝑟crelexp 14240  t*reccrtrcl 14275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-mulcl 10397  ax-i2m1 10403
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-nn 11440  df-n0 11708  df-relexp 14241  df-rtrclrec 14276
This theorem is referenced by:  dfrtrcl2  14282
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