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Theorem reldmmpt2 5554
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 = (x A, y B𝐶)
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 5531 . 2 Rel dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
2 rngop.1 . . . . 5 𝐹 = (x A, y B𝐶)
3 df-mpt2 5460 . . . . 5 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
42, 3eqtri 2057 . . . 4 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
54dmeqi 4479 . . 3 dom 𝐹 = dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
65releqi 4366 . 2 (Rel dom 𝐹 ↔ Rel dom {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)})
71, 6mpbir 134 1 Rel dom 𝐹
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  dom cdm 4288  Rel wrel 4293  {coprab 5456  cmpt2 5457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-oprab 5459  df-mpt2 5460
This theorem is referenced by: (None)
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