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Theorem relexp0g 14916
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 2734 . . 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 772 . . . . 5 ((𝑅𝑉 ∧ (𝑟 = 𝑅𝑛 = 0)) → 𝑛 = 0)
32iftrued 4498 . . . 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 5863 . . . . . . 7 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
5 rneq 5895 . . . . . . 7 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
64, 5uneq12d 4128 . . . . . 6 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
76reseq2d 5941 . . . . 5 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
87ad2antrl 727 . . . 4 ((𝑅𝑉 ∧ (𝑟 = 𝑅𝑛 = 0)) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
93, 8eqtrd 2773 . . 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 3465 . . 3 (𝑅𝑉𝑅 ∈ V)
11 0nn0 12436 . . . 4 0 ∈ ℕ0
1211a1i 11 . . 3 (𝑅𝑉 → 0 ∈ ℕ0)
13 dmexg 7844 . . . . 5 (𝑅𝑉 → dom 𝑅 ∈ V)
14 rnexg 7845 . . . . 5 (𝑅𝑉 → ran 𝑅 ∈ V)
15 unexg 7687 . . . . 5 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
1613, 14, 15syl2anc 585 . . . 4 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
17 resiexg 7855 . . . 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 7511 . 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 14914 . . 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 7367 . . . . 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 2735 . . . 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 341 . . 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 230 1 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cun 3912  ifcif 4490  cmpt 5192   I cid 5534  dom cdm 5637  ran crn 5638  cres 5639  ccom 5641  cfv 6500  (class class class)co 7361  cmpo 7363  0cc0 11059  1c1 11060  0cn0 12421  seqcseq 13915  𝑟crelexp 14913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-mulcl 11121  ax-i2m1 11127
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-n0 12422  df-relexp 14914
This theorem is referenced by:  relexp0  14917  relexpcnv  14929  relexp0rel  14931  relexpdmg  14936  relexprng  14940  relexpfld  14943  relexpaddg  14947  dfrcl3  42039  fvmptiunrelexplb0d  42048  brfvrcld2  42056  relexp0eq  42065  iunrelexp0  42066  relexpiidm  42068  relexpss1d  42069  relexpmulg  42074  iunrelexpmin2  42076  relexp01min  42077  relexp0a  42080  relexpxpmin  42081  relexpaddss  42082  dfrtrcl3  42097  cotrclrcl  42106
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