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Theorem relssdmrn 5209
Description: A relation is included in the cross 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 19 . 2  |-  ( Rel 
A  ->  Rel  A )
2 19.8a 1730 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. y <. x ,  y >.  e.  A )
3 19.8a 1730 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. x <. x ,  y >.  e.  A )
4 opelxp 4735 . . . . 5  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( x  e.  dom  A  /\  y  e.  ran  A ) )
5 vex 2804 . . . . . . 7  |-  x  e. 
_V
65eldm2 4893 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
7 vex 2804 . . . . . . 7  |-  y  e. 
_V
87elrn2 4934 . . . . . 6  |-  ( y  e.  ran  A  <->  E. x <. x ,  y >.  e.  A )
96, 8anbi12i 678 . . . . 5  |-  ( ( x  e.  dom  A  /\  y  e.  ran  A )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. x <. x ,  y >.  e.  A ) )
104, 9bitri 240 . . . 4  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. x <. x ,  y
>.  e.  A ) )
112, 3, 10sylanbrc 645 . . 3  |-  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  ( dom  A  X.  ran  A ) )
1211a1i 10 . 2  |-  ( Rel 
A  ->  ( <. x ,  y >.  e.  A  -> 
<. x ,  y >.  e.  ( dom  A  X.  ran  A ) ) )
131, 12relssdv 4795 1  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    e. wcel 1696    C_ wss 3165   <.cop 3656    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710
This theorem is referenced by:  cnvssrndm  5210  cossxp  5211  relrelss  5212  relfld  5214  cnvexg  5224  coexg  5231  fssxp  5416  resfunexgALT  5754  cofunexg  5755  fnexALT  5758  oprabss  5949  erssxp  6699  wunco  8371  imasless  13458  sylow2a  14946  gsum2d  15239  znleval  16524  tsmsxp  17853  oprabex2gpop  25139  relinccppr  25232  prismorcsetlem  26015  prismorcset  26017  prismorcsetlemc  26020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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