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Theorem dmcosseq 4946
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 4944 . . 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 3174 . . . . . . . 8  |-  ( ran 
B  C_  dom  A  -> 
( y  e.  ran  B  ->  y  e.  dom  A ) )
4 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
54elrn 4919 . . . . . . . . . 10  |-  ( y  e.  ran  B  <->  E. x  x B y )
64eldm 4876 . . . . . . . . . 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 1718 . . . . . . . . . . 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 1608 . . . . . . . . . 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 1608 . . . . . 6  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. y E. z ( x B y  /\  y A z ) ) )
16 excom 1786 . . . . . 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 2791 . . . . . . 7  |-  x  e. 
_V
19 vex 2791 . . . . . . 7  |-  z  e. 
_V
2018, 19opelco 4853 . . . . . 6  |-  ( <.
x ,  z >.  e.  ( A  o.  B
)  <->  E. y ( x B y  /\  y A z ) )
2120exbii 1569 . . . . 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 4876 . . . 4  |-  ( x  e.  dom  B  <->  E. y  x B y )
2418eldm2 4877 . . . 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 3185 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  B  C_  dom  ( A  o.  B ) )
272, 26eqssd 3196 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 1528    = wceq 1623    e. wcel 1684    C_ wss 3152   <.cop 3643   class class class wbr 4023   dom cdm 4689   ran crn 4690    o. ccom 4693
This theorem is referenced by:  dmcoeq  4947  fnco  5352  fnresfnco  27989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700
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