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Theorem reldmmpt2 5770
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  e.  A ,  y  e.  B  |->  C )
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 5747 . 2  |-  Rel  dom  {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
2 rngop.1 . . . . 5  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
3 df-mpt2 5671 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
42, 3eqtri 2109 . . . 4  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
54dmeqi 4650 . . 3  |-  dom  F  =  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
65releqi 4534 . 2  |-  ( Rel 
dom  F  <->  Rel  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) } )
71, 6mpbir 145 1  |-  Rel  dom  F
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1290    e. wcel 1439   dom cdm 4452   Rel wrel 4457   {coprab 5667    |-> cmpt2 5668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-dm 4462  df-oprab 5670  df-mpt2 5671
This theorem is referenced by:  reldmmap  6428  reldmsets  11584  reldmress  11613  reldmprds  11739
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