MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relssdmrn Structured version   Unicode version

Theorem relssdmrn 5436
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 1865 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. y <. x ,  y >.  e.  A )
3 19.8a 1865 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. x <. x ,  y >.  e.  A )
4 opelxp 4943 . . . . 5  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( x  e.  dom  A  /\  y  e.  ran  A ) )
5 vex 3037 . . . . . . 7  |-  x  e. 
_V
65eldm2 5114 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
7 vex 3037 . . . . . . 7  |-  y  e. 
_V
87elrn2 5155 . . . . . 6  |-  ( y  e.  ran  A  <->  E. x <. x ,  y >.  e.  A )
96, 8anbi12i 695 . . . . 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 662 . . 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 5008 1  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1620    e. wcel 1826    C_ wss 3389   <.cop 3950    X. cxp 4911   dom cdm 4913   ran crn 4914   Rel wrel 4918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924
This theorem is referenced by:  cnvssrndm  5437  cossxp  5438  relrelss  5439  relfld  5441  fssxp  5651  oprabss  6287  cnvexg  6645  resfunexgALT  6662  cofunexg  6663  fnexALT  6665  erssxp  7252  wunco  9022  trclublem  12833  trclubi  12834  trclub  12836  reltrclfv  12855  imasless  14947  sylow2a  16756  gsum2d  17113  gsum2dOLD  17114  znleval  18684  tsmsxp  20742  relfi  27592  idssxp  27607  fcnvgreu  27660  trrelsuperreldg  38208  rp-imass  38265  idhe  38282
  Copyright terms: Public domain W3C validator