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Theorem dmcosseq 4857
Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcosseq  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)

Proof of Theorem dmcosseq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 4855 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
21a1i 9 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  C_  dom  B )
3 ssel 3122 . . . . . . . 8  |-  ( ran 
B  C_  dom  A  -> 
( y  e.  ran  B  ->  y  e.  dom  A ) )
4 vex 2715 . . . . . . . . . . 11  |-  y  e. 
_V
54elrn 4829 . . . . . . . . . 10  |-  ( y  e.  ran  B  <->  E. x  x B y )
64eldm 4783 . . . . . . . . . 10  |-  ( y  e.  dom  A  <->  E. z 
y A z )
75, 6imbi12i 238 . . . . . . . . 9  |-  ( ( y  e.  ran  B  ->  y  e.  dom  A
)  <->  ( E. x  x B y  ->  E. z 
y A z ) )
8 19.8a 1570 . . . . . . . . . . 11  |-  ( x B y  ->  E. x  x B y )
98imim1i 60 . . . . . . . . . 10  |-  ( ( E. x  x B y  ->  E. z 
y A z )  ->  ( x B y  ->  E. z 
y A z ) )
10 pm3.2 138 . . . . . . . . . . 11  |-  ( x B y  ->  (
y A z  -> 
( x B y  /\  y A z ) ) )
1110eximdv 1860 . . . . . . . . . 10  |-  ( x B y  ->  ( E. z  y A
z  ->  E. z
( x B y  /\  y A z ) ) )
129, 11sylcom 28 . . . . . . . . 9  |-  ( ( E. x  x B y  ->  E. z 
y A z )  ->  ( x B y  ->  E. z
( x B y  /\  y A z ) ) )
137, 12sylbi 120 . . . . . . . 8  |-  ( ( y  e.  ran  B  ->  y  e.  dom  A
)  ->  ( x B y  ->  E. z
( x B y  /\  y A z ) ) )
143, 13syl 14 . . . . . . 7  |-  ( ran 
B  C_  dom  A  -> 
( x B y  ->  E. z ( x B y  /\  y A z ) ) )
1514eximdv 1860 . . . . . 6  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. y E. z ( x B y  /\  y A z ) ) )
16 excom 1644 . . . . . 6  |-  ( E. z E. y ( x B y  /\  y A z )  <->  E. y E. z ( x B y  /\  y A z ) )
1715, 16syl6ibr 161 . . . . 5  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. z E. y ( x B y  /\  y A z ) ) )
18 vex 2715 . . . . . . 7  |-  x  e. 
_V
19 vex 2715 . . . . . . 7  |-  z  e. 
_V
2018, 19opelco 4758 . . . . . 6  |-  ( <.
x ,  z >.  e.  ( A  o.  B
)  <->  E. y ( x B y  /\  y A z ) )
2120exbii 1585 . . . . 5  |-  ( E. z <. x ,  z
>.  e.  ( A  o.  B )  <->  E. z E. y ( x B y  /\  y A z ) )
2217, 21syl6ibr 161 . . . 4  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. z <. x ,  z >.  e.  ( A  o.  B
) ) )
2318eldm 4783 . . . 4  |-  ( x  e.  dom  B  <->  E. y  x B y )
2418eldm2 4784 . . . 4  |-  ( x  e.  dom  ( A  o.  B )  <->  E. z <. x ,  z >.  e.  ( A  o.  B
) )
2522, 23, 243imtr4g 204 . . 3  |-  ( ran 
B  C_  dom  A  -> 
( x  e.  dom  B  ->  x  e.  dom  ( A  o.  B
) ) )
2625ssrdv 3134 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  B  C_  dom  ( A  o.  B ) )
272, 26eqssd 3145 1  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335   E.wex 1472    e. wcel 2128    C_ wss 3102   <.cop 3563   class class class wbr 3965   dom cdm 4586   ran crn 4587    o. ccom 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597
This theorem is referenced by:  dmcoeq  4858  fnco  5278
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