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Theorem relexp0g 14969
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 4537 . . . 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 5904 . . . . . . 7 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
5 rneq 5936 . . . . . . 7 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
64, 5uneq12d 4165 . . . . . 6 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
76reseq2d 5982 . . . . 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 3493 . . 3 (𝑅𝑉𝑅 ∈ V)
11 0nn0 12487 . . . 4 0 ∈ ℕ0
1211a1i 11 . . 3 (𝑅𝑉 → 0 ∈ ℕ0)
13 dmexg 7894 . . . . 5 (𝑅𝑉 → dom 𝑅 ∈ V)
14 rnexg 7895 . . . . 5 (𝑅𝑉 → ran 𝑅 ∈ V)
15 unexg 7736 . . . . 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 7905 . . . 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 7560 . 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 14967 . . 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 7415 . . . . 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 3475  cun 3947  ifcif 4529  cmpt 5232   I cid 5574  dom cdm 5677  ran crn 5678  cres 5679  ccom 5681  cfv 6544  (class class class)co 7409  cmpo 7411  0cc0 11110  1c1 11111  0cn0 12472  seqcseq 13966  𝑟crelexp 14966
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-mulcl 11172  ax-i2m1 11178
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-n0 12473  df-relexp 14967
This theorem is referenced by:  relexp0  14970  relexpcnv  14982  relexp0rel  14984  relexpdmg  14989  relexprng  14993  relexpfld  14996  relexpaddg  15000  dfrcl3  42426  fvmptiunrelexplb0d  42435  brfvrcld2  42443  relexp0eq  42452  iunrelexp0  42453  relexpiidm  42455  relexpss1d  42456  relexpmulg  42461  iunrelexpmin2  42463  relexp01min  42464  relexp0a  42467  relexpxpmin  42468  relexpaddss  42469  dfrtrcl3  42484  cotrclrcl  42493
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