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Theorem relssdmrn 5355
Description: A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
relssdmrn  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )

Proof of Theorem relssdmrn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2  |-  ( Rel 
A  ->  Rel  A )
2 19.8a 1798 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. y <. x ,  y >.  e.  A )
3 19.8a 1798 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. x <. x ,  y >.  e.  A )
4 opelxp 4865 . . . . 5  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( x  e.  dom  A  /\  y  e.  ran  A ) )
5 vex 2973 . . . . . . 7  |-  x  e. 
_V
65eldm2 5034 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
7 vex 2973 . . . . . . 7  |-  y  e. 
_V
87elrn2 5075 . . . . . 6  |-  ( y  e.  ran  A  <->  E. x <. x ,  y >.  e.  A )
96, 8anbi12i 692 . . . . 5  |-  ( ( x  e.  dom  A  /\  y  e.  ran  A )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. x <. x ,  y >.  e.  A ) )
104, 9bitri 249 . . . 4  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. x <. x ,  y
>.  e.  A ) )
112, 3, 10sylanbrc 659 . . 3  |-  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  ( dom  A  X.  ran  A ) )
1211a1i 11 . 2  |-  ( Rel 
A  ->  ( <. x ,  y >.  e.  A  -> 
<. x ,  y >.  e.  ( dom  A  X.  ran  A ) ) )
131, 12relssdv 4928 1  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1591    e. wcel 1761    C_ wss 3325   <.cop 3880    X. cxp 4834   dom cdm 4836   ran crn 4837   Rel wrel 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847
This theorem is referenced by:  cnvssrndm  5356  cossxp  5357  relrelss  5358  relfld  5360  fssxp  5567  oprabss  6175  cnvexg  6523  resfunexgALT  6539  cofunexg  6540  fnexALT  6542  erssxp  7120  wunco  8896  imasless  14474  sylow2a  16111  gsum2d  16453  gsum2dOLD  16454  znleval  17946  tsmsxp  19688  relfi  25875  fcnvgreu  25926
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