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Theorem relexp0g 15061
Description: A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
Assertion
Ref Expression
relexp0g (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))

Proof of Theorem relexp0g
Dummy variables 𝑛 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2738 . . 3 (𝑅𝑉 → (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))))
2 simprr 773 . . . . 5 ((𝑅𝑉 ∧ (𝑟 = 𝑅𝑛 = 0)) → 𝑛 = 0)
32iftrued 4533 . . . 4 ((𝑅𝑉 ∧ (𝑟 = 𝑅𝑛 = 0)) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
4 dmeq 5914 . . . . . . 7 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
5 rneq 5947 . . . . . . 7 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
64, 5uneq12d 4169 . . . . . 6 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
76reseq2d 5997 . . . . 5 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
87ad2antrl 728 . . . 4 ((𝑅𝑉 ∧ (𝑟 = 𝑅𝑛 = 0)) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
93, 8eqtrd 2777 . . 3 ((𝑅𝑉 ∧ (𝑟 = 𝑅𝑛 = 0)) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
10 elex 3501 . . 3 (𝑅𝑉𝑅 ∈ V)
11 0nn0 12541 . . . 4 0 ∈ ℕ0
1211a1i 11 . . 3 (𝑅𝑉 → 0 ∈ ℕ0)
13 dmexg 7923 . . . . 5 (𝑅𝑉 → dom 𝑅 ∈ V)
14 rnexg 7924 . . . . 5 (𝑅𝑉 → ran 𝑅 ∈ V)
15 unexg 7763 . . . . 5 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
1613, 14, 15syl2anc 584 . . . 4 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
17 resiexg 7934 . . . 4 ((dom 𝑅 ∪ ran 𝑅) ∈ V → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
1816, 17syl 17 . . 3 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
191, 9, 10, 12, 18ovmpod 7585 . 2 (𝑅𝑉 → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
20 df-relexp 15059 . . 3 𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
21 oveq 7437 . . . . 5 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (𝑅𝑟0) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))0))
2221eqeq1d 2739 . . . 4 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → ((𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
2322imbi2d 340 . . 3 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → ((𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ↔ (𝑅𝑉 → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
2420, 23ax-mp 5 . 2 ((𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ↔ (𝑅𝑉 → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
2519, 24mpbir 231 1 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  ifcif 4525  cmpt 5225   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156  0cn0 12526  seqcseq 14042  𝑟crelexp 15058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-n0 12527  df-relexp 15059
This theorem is referenced by:  relexp0  15062  relexpcnv  15074  relexp0rel  15076  relexpdmg  15081  relexprng  15085  relexpfld  15088  relexpaddg  15092  dfrcl3  43688  fvmptiunrelexplb0d  43697  brfvrcld2  43705  relexp0eq  43714  iunrelexp0  43715  relexpiidm  43717  relexpss1d  43718  relexpmulg  43723  iunrelexpmin2  43725  relexp01min  43726  relexp0a  43729  relexpxpmin  43730  relexpaddss  43731  dfrtrcl3  43746  cotrclrcl  43755
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