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Theorem reldmmpo 7567
Description: The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
reldmmpo Rel dom 𝐹
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem reldmmpo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 reldmoprab 7540 . 2 Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
3 df-mpo 7436 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
42, 3eqtri 2765 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
54dmeqi 5915 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
65releqi 5787 . 2 (Rel dom 𝐹 ↔ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
71, 6mpbir 231 1 Rel dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  dom cdm 5685  Rel wrel 5690  {coprab 7432  cmpo 7433
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-pr 5432
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-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  reldmmap  8875  reldmrelexp  15060  reldmsets  17202  reldmress  17276  reldmprds  17493  gsum0  18697  reldmghm  19232  oppglsm  19660  reldmdprd  20017  reldmlmhm  21024  zrhval  21518  reldmdsmm  21753  frlmrcl  21777  reldmpsr  21934  reldmmpl  22008  reldmopsr  22063  reldmevls  22108  reldmmhp  22141  vr1val  22193  reldmevls1  22321  evl1fval  22332  matbas0pc  22413  mdetfval  22592  madufval  22643  qtopres  23706  fgabs  23887  reldmtng  24651  reldmnghm  24733  reldmnmhm  24734  dvbsss  25937  reldmmdeg  26096  nbgrprc0  29351  wwlksn  29857  of0r  32688  reldmrloc  33261  erlval  33262  reldmresv  33352  bj-restsnid  37088  mzpmfp  42758  brovmptimex  44040  clnbgrprc0  47807  grimdmrel  47866  grlimdmrel  47947  1aryenef  48566  2aryenef  48577  reldmup  48932  upfval  48933  reldmup2  48938  reldmxpcALT  48953  fucofvalne  49020
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