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

Theorem reldmmpt2 6725
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 6699 . 2 Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
3 df-mpt2 6610 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
42, 3eqtri 2648 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
54dmeqi 5290 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
65releqi 5168 . 2 (Rel dom 𝐹 ↔ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
71, 6mpbir 221 1 Rel dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1992  dom cdm 5079  Rel wrel 5084  {coprab 6606  cmpt2 6607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-dm 5089  df-oprab 6609  df-mpt2 6610
This theorem is referenced by:  reldmmap  7812  reldmsets  15802  reldmress  15842  reldmprds  16025  gsum0  17194  reldmghm  17575  oppglsm  17973  reldmdprd  18312  reldmlmhm  18939  reldmpsr  19275  reldmmpl  19341  reldmopsr  19387  reldmevls  19431  vr1val  19476  reldmevls1  19596  evl1fval  19606  zrhval  19770  reldmdsmm  19991  frlmrcl  20015  matbas0pc  20129  mdetfval  20306  madufval  20357  qtopres  21406  fgabs  21588  reldmtng  22347  reldmnghm  22421  reldmnmhm  22422  dvbsss  23567  reldmmdeg  23716  nbgrprc0  26110  wwlksn  26592  wwlks2onv  26710  clwwlksn  26742  reldmresv  29603  bj-restsnid  32669  mzpmfp  36776  brovmptimex  37793
  Copyright terms: Public domain W3C validator