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Theorem dmcoss 4944
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 1706 . . . 4  |-  F/ y E. y  x B y
2 exsimpl 1579 . . . . 5  |-  ( E. z ( x B z  /\  z A y )  ->  E. z  x B z )
3 vex 2791 . . . . . 6  |-  x  e. 
_V
4 vex 2791 . . . . . 6  |-  y  e. 
_V
53, 4opelco 4853 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  <->  E. z ( x B z  /\  z A y ) )
6 breq2 4027 . . . . . 6  |-  ( y  =  z  ->  (
x B y  <->  x B
z ) )
76cbvexv 1943 . . . . 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 1801 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  o.  B )  ->  E. y  x B y )
103eldm2 4877 . . 3  |-  ( x  e.  dom  ( A  o.  B )  <->  E. y <. x ,  y >.  e.  ( A  o.  B
) )
113eldm 4876 . . 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 3184 1  |-  dom  ( A  o.  B )  C_ 
dom  B
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    e. wcel 1684    C_ wss 3152   <.cop 3643   class class class wbr 4023   dom cdm 4689    o. ccom 4693
This theorem is referenced by:  rncoss  4945  dmcosseq  4946  cossxp  5195  coexg  5215  fvco4i  5597  cofunexg  5739  fin23lem30  7968  wunco  8355  znleval  16508  tngtopn  18166  xppreima  23211  relexpdm  24032  mvdco  27388  f1omvdconj  27389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-co 4698  df-dm 4699
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