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Theorem reldmmpt2 6669
 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 6643 . 2 Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
3 df-mpt2 6554 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
42, 3eqtri 2632 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
54dmeqi 5247 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
65releqi 5125 . 2 (Rel dom 𝐹 ↔ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
71, 6mpbir 220 1 Rel dom 𝐹
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  dom cdm 5038  Rel wrel 5043  {coprab 6550   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  reldmmap  7753  reldmsets  15718  reldmress  15753  reldmprds  15932  gsum0  17101  reldmghm  17482  oppglsm  17880  reldmdprd  18219  reldmlmhm  18846  reldmpsr  19182  reldmmpl  19248  reldmopsr  19294  reldmevls  19338  vr1val  19383  reldmevls1  19503  evl1fval  19513  zrhval  19675  reldmdsmm  19896  frlmrcl  19920  matbas0pc  20034  mdetfval  20211  madufval  20262  qtopres  21311  fgabs  21493  reldmtng  22252  reldmnghm  22326  reldmnmhm  22327  dvbsss  23472  reldmmdeg  23621  reldmresv  29157  bj-restsnid  32221  mzpmfp  36328  brovmptimex  37345  nbgrprc0  40555  wwlksn  41040  wwlks2onv  41158  clwwlksn  41189
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