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Theorem relssdmrn 5149
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 1590 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. y <. x ,  y >.  e.  A )
3 19.8a 1590 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. x <. x ,  y >.  e.  A )
4 opelxp 4656 . . . . 5  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( x  e.  dom  A  /\  y  e.  ran  A ) )
5 vex 2740 . . . . . . 7  |-  x  e. 
_V
65eldm2 4825 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
7 vex 2740 . . . . . . 7  |-  y  e. 
_V
87elrn2 4869 . . . . . 6  |-  ( y  e.  ran  A  <->  E. x <. x ,  y >.  e.  A )
96, 8anbi12i 460 . . . . 5  |-  ( ( x  e.  dom  A  /\  y  e.  ran  A )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. x <. x ,  y >.  e.  A ) )
104, 9bitri 184 . . . 4  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. x <. x ,  y
>.  e.  A ) )
112, 3, 10sylanbrc 417 . . 3  |-  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  ( dom  A  X.  ran  A ) )
1211a1i 9 . 2  |-  ( Rel 
A  ->  ( <. x ,  y >.  e.  A  -> 
<. x ,  y >.  e.  ( dom  A  X.  ran  A ) ) )
131, 12relssdv 4718 1  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1492    e. wcel 2148    C_ wss 3129   <.cop 3595    X. cxp 4624   dom cdm 4626   ran crn 4627   Rel wrel 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637
This theorem is referenced by:  cnvssrndm  5150  cossxp  5151  relrelss  5155  relfld  5157  cnvexg  5166  fssxp  5383  oprabss  5960  resfunexgALT  6108  cofunexg  6109  fnexALT  6111  funexw  6112  erssxp  6557
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