MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldmmpt2 Structured version   Visualization version   GIF version

Theorem reldmmpt2 7048
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
reldmmpt2 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 reldmmpt2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 reldmoprab 7022 . 2 Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
3 df-mpt2 6927 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
42, 3eqtri 2801 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
54dmeqi 5570 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
65releqi 5450 . 2 (Rel dom 𝐹 ↔ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
71, 6mpbir 223 1 Rel dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wcel 2106  dom cdm 5355  Rel wrel 5360  {coprab 6923  cmpt2 6924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-dm 5365  df-oprab 6926  df-mpt2 6927
This theorem is referenced by:  reldmmap  8149  reldmsets  16283  reldmress  16322  reldmprds  16495  gsum0  17664  reldmghm  18043  oppglsm  18441  reldmdprd  18783  reldmlmhm  19420  reldmpsr  19758  reldmmpl  19824  reldmopsr  19870  reldmevls  19913  vr1val  19958  reldmevls1  20078  evl1fval  20088  zrhval  20252  reldmdsmm  20476  frlmrcl  20500  matbas0pc  20619  mdetfval  20797  madufval  20848  qtopres  21910  fgabs  22091  reldmtng  22850  reldmnghm  22924  reldmnmhm  22925  dvbsss  24103  reldmmdeg  24254  nbgrprc0  26681  wwlksn  27186  reldmresv  30402  bj-restsnid  33627  mzpmfp  38262  brovmptimex  39273
  Copyright terms: Public domain W3C validator