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Theorem dmcoss 4946
Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcoss  |-  dom  ( A  o.  B )  C_ 
dom  B

Proof of Theorem dmcoss
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfe1 1708 . . . 4  |-  F/ y E. y  x B y
2 exsimpl 1581 . . . . 5  |-  ( E. z ( x B z  /\  z A y )  ->  E. z  x B z )
3 vex 2793 . . . . . 6  |-  x  e. 
_V
4 vex 2793 . . . . . 6  |-  y  e. 
_V
53, 4opelco 4855 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  <->  E. z ( x B z  /\  z A y ) )
6 breq2 4029 . . . . . 6  |-  ( y  =  z  ->  (
x B y  <->  x B
z ) )
76cbvexv 1945 . . . . 5  |-  ( E. y  x B y  <->  E. z  x B
z )
82, 5, 73imtr4i 257 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  ->  E. y  x B y )
91, 8exlimi 1803 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  o.  B )  ->  E. y  x B y )
103eldm2 4879 . . 3  |-  ( x  e.  dom  ( A  o.  B )  <->  E. y <. x ,  y >.  e.  ( A  o.  B
) )
113eldm 4878 . . 3  |-  ( x  e.  dom  B  <->  E. y  x B y )
129, 10, 113imtr4i 257 . 2  |-  ( x  e.  dom  ( A  o.  B )  ->  x  e.  dom  B )
1312ssriv 3186 1  |-  dom  ( A  o.  B )  C_ 
dom  B
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1530    e. wcel 1686    C_ wss 3154   <.cop 3645   class class class wbr 4025   dom cdm 4691    o. ccom 4695
This theorem is referenced by:  rncoss  4947  dmcosseq  4948  cossxp  5197  coexg  5217  fvco4i  5599  cofunexg  5741  fin23lem30  7970  wunco  8357  znleval  16510  tngtopn  18168  xppreima  23213  relexpdm  24034  mvdco  27399  f1omvdconj  27400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-co 4700  df-dm 4701
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