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Theorem dmcoss 4650
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 1426 . . . 4  |-  F/ y E. y  x B y
2 exsimpl 1549 . . . . 5  |-  ( E. z ( x B z  /\  z A y )  ->  E. z  x B z )
3 vex 2613 . . . . . 6  |-  x  e. 
_V
4 vex 2613 . . . . . 6  |-  y  e. 
_V
53, 4opelco 4556 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  <->  E. z ( x B z  /\  z A y ) )
6 breq2 3810 . . . . . 6  |-  ( y  =  z  ->  (
x B y  <->  x B
z ) )
76cbvexv 1838 . . . . 5  |-  ( E. y  x B y  <->  E. z  x B
z )
82, 5, 73imtr4i 199 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  ->  E. y  x B y )
91, 8exlimi 1526 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  o.  B )  ->  E. y  x B y )
103eldm2 4582 . . 3  |-  ( x  e.  dom  ( A  o.  B )  <->  E. y <. x ,  y >.  e.  ( A  o.  B
) )
113eldm 4581 . . 3  |-  ( x  e.  dom  B  <->  E. y  x B y )
129, 10, 113imtr4i 199 . 2  |-  ( x  e.  dom  ( A  o.  B )  ->  x  e.  dom  B )
1312ssriv 3013 1  |-  dom  ( A  o.  B )  C_ 
dom  B
Colors of variables: wff set class
Syntax hints:    /\ wa 102   E.wex 1422    e. wcel 1434    C_ wss 2983   <.cop 3420   class class class wbr 3806   dom cdm 4392    o. ccom 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-co 4401  df-dm 4402
This theorem is referenced by:  rncoss  4651  dmcosseq  4652  cossxp  4894  funco  4991  cofunexg  5790
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