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Theorem dmcoss 4960
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 1718 . . . 4  |-  F/ y E. y  x B y
2 exsimpl 1582 . . . . 5  |-  ( E. z ( x B z  /\  z A y )  ->  E. z  x B z )
3 vex 2804 . . . . . 6  |-  x  e. 
_V
4 vex 2804 . . . . . 6  |-  y  e. 
_V
53, 4opelco 4869 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  <->  E. z ( x B z  /\  z A y ) )
6 breq2 4043 . . . . . 6  |-  ( y  =  z  ->  (
x B y  <->  x B
z ) )
76cbvexv 1956 . . . . 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 1813 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  o.  B )  ->  E. y  x B y )
103eldm2 4893 . . 3  |-  ( x  e.  dom  ( A  o.  B )  <->  E. y <. x ,  y >.  e.  ( A  o.  B
) )
113eldm 4892 . . 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 3197 1  |-  dom  ( A  o.  B )  C_ 
dom  B
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    e. wcel 1696    C_ wss 3165   <.cop 3656   class class class wbr 4039   dom cdm 4705    o. ccom 4709
This theorem is referenced by:  rncoss  4961  dmcosseq  4962  cossxp  5211  coexg  5231  fvco4i  5613  cofunexg  5755  fin23lem30  7984  wunco  8371  znleval  16524  tngtopn  18182  xppreima  23226  relexpdm  24047  mvdco  27491  f1omvdconj  27492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-co 4714  df-dm 4715
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