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

Theorem relssdmrn 5381
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 20 . 2  |-  ( Rel 
A  ->  Rel  A )
2 19.8a 1762 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. y <. x ,  y >.  e.  A )
3 19.8a 1762 . . . 4  |-  ( <.
x ,  y >.  e.  A  ->  E. x <. x ,  y >.  e.  A )
4 opelxp 4899 . . . . 5  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( x  e.  dom  A  /\  y  e.  ran  A ) )
5 vex 2951 . . . . . . 7  |-  x  e. 
_V
65eldm2 5059 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
7 vex 2951 . . . . . . 7  |-  y  e. 
_V
87elrn2 5100 . . . . . 6  |-  ( y  e.  ran  A  <->  E. x <. x ,  y >.  e.  A )
96, 8anbi12i 679 . . . . 5  |-  ( ( x  e.  dom  A  /\  y  e.  ran  A )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. x <. x ,  y >.  e.  A ) )
104, 9bitri 241 . . . 4  |-  ( <.
x ,  y >.  e.  ( dom  A  X.  ran  A )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. x <. x ,  y
>.  e.  A ) )
112, 3, 10sylanbrc 646 . . 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 4959 1  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725    C_ wss 3312   <.cop 3809    X. cxp 4867   dom cdm 4869   ran crn 4870   Rel wrel 4874
This theorem is referenced by:  cnvssrndm  5382  cossxp  5383  relrelss  5384  relfld  5386  cnvexg  5396  fssxp  5593  resfunexgALT  5949  cofunexg  5950  fnexALT  5953  oprabss  6150  erssxp  6919  wunco  8597  imasless  13753  sylow2a  15241  gsum2d  15534  znleval  16823  tsmsxp  18172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-dm 4879  df-rn 4880
  Copyright terms: Public domain W3C validator