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Theorem dmcosseq 4983
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 4981 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
21a1i 10 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  C_  dom  B )
3 ssel 3208 . . . . . . . 8  |-  ( ran 
B  C_  dom  A  -> 
( y  e.  ran  B  ->  y  e.  dom  A ) )
4 vex 2825 . . . . . . . . . . 11  |-  y  e. 
_V
54elrn 4956 . . . . . . . . . 10  |-  ( y  e.  ran  B  <->  E. x  x B y )
64eldm 4913 . . . . . . . . . 10  |-  ( y  e.  dom  A  <->  E. z 
y A z )
75, 6imbi12i 316 . . . . . . . . 9  |-  ( ( y  e.  ran  B  ->  y  e.  dom  A
)  <->  ( E. x  x B y  ->  E. z 
y A z ) )
8 19.8a 1739 . . . . . . . . . . 11  |-  ( x B y  ->  E. x  x B y )
98imim1i 54 . . . . . . . . . 10  |-  ( ( E. x  x B y  ->  E. z 
y A z )  ->  ( x B y  ->  E. z 
y A z ) )
10 pm3.2 434 . . . . . . . . . . 11  |-  ( x B y  ->  (
y A z  -> 
( x B y  /\  y A z ) ) )
1110eximdv 1613 . . . . . . . . . 10  |-  ( x B y  ->  ( E. z  y A
z  ->  E. z
( x B y  /\  y A z ) ) )
129, 11sylcom 25 . . . . . . . . 9  |-  ( ( E. x  x B y  ->  E. z 
y A z )  ->  ( x B y  ->  E. z
( x B y  /\  y A z ) ) )
137, 12sylbi 187 . . . . . . . 8  |-  ( ( y  e.  ran  B  ->  y  e.  dom  A
)  ->  ( x B y  ->  E. z
( x B y  /\  y A z ) ) )
143, 13syl 15 . . . . . . 7  |-  ( ran 
B  C_  dom  A  -> 
( x B y  ->  E. z ( x B y  /\  y A z ) ) )
1514eximdv 1613 . . . . . 6  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. y E. z ( x B y  /\  y A z ) ) )
16 excom 1817 . . . . . 6  |-  ( E. z E. y ( x B y  /\  y A z )  <->  E. y E. z ( x B y  /\  y A z ) )
1715, 16syl6ibr 218 . . . . 5  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. z E. y ( x B y  /\  y A z ) ) )
18 vex 2825 . . . . . . 7  |-  x  e. 
_V
19 vex 2825 . . . . . . 7  |-  z  e. 
_V
2018, 19opelco 4890 . . . . . 6  |-  ( <.
x ,  z >.  e.  ( A  o.  B
)  <->  E. y ( x B y  /\  y A z ) )
2120exbii 1573 . . . . 5  |-  ( E. z <. x ,  z
>.  e.  ( A  o.  B )  <->  E. z E. y ( x B y  /\  y A z ) )
2217, 21syl6ibr 218 . . . 4  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. z <. x ,  z >.  e.  ( A  o.  B
) ) )
2318eldm 4913 . . . 4  |-  ( x  e.  dom  B  <->  E. y  x B y )
2418eldm2 4914 . . . 4  |-  ( x  e.  dom  ( A  o.  B )  <->  E. z <. x ,  z >.  e.  ( A  o.  B
) )
2522, 23, 243imtr4g 261 . . 3  |-  ( ran 
B  C_  dom  A  -> 
( x  e.  dom  B  ->  x  e.  dom  ( A  o.  B
) ) )
2625ssrdv 3219 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  B  C_  dom  ( A  o.  B ) )
272, 26eqssd 3230 1  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701    C_ wss 3186   <.cop 3677   class class class wbr 4060   dom cdm 4726   ran crn 4727    o. ccom 4730
This theorem is referenced by:  dmcoeq  4984  fnco  5389  fnresfnco  27139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737
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