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Theorem dmcoss 4672
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 1428 . . . 4  |-  F/ y E. y  x B y
2 exsimpl 1551 . . . . 5  |-  ( E. z ( x B z  /\  z A y )  ->  E. z  x B z )
3 vex 2618 . . . . . 6  |-  x  e. 
_V
4 vex 2618 . . . . . 6  |-  y  e. 
_V
53, 4opelco 4578 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  B
)  <->  E. z ( x B z  /\  z A y ) )
6 breq2 3826 . . . . . 6  |-  ( y  =  z  ->  (
x B y  <->  x B
z ) )
76cbvexv 1840 . . . . 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 1528 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  o.  B )  ->  E. y  x B y )
103eldm2 4604 . . 3  |-  ( x  e.  dom  ( A  o.  B )  <->  E. y <. x ,  y >.  e.  ( A  o.  B
) )
113eldm 4603 . . 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 3018 1  |-  dom  ( A  o.  B )  C_ 
dom  B
Colors of variables: wff set class
Syntax hints:    /\ wa 102   E.wex 1424    e. wcel 1436    C_ wss 2988   <.cop 3434   class class class wbr 3822   dom cdm 4413    o. ccom 4417
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-co 4422  df-dm 4423
This theorem is referenced by:  rncoss  4673  dmcosseq  4674  cossxp  4921  funco  5021  cofunexg  5841  casefun  6723  djufun  6731
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