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Theorem dmcosseq 4631
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 4629 . . 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 2967 . . . . . . . 8  |-  ( ran 
B  C_  dom  A  -> 
( y  e.  ran  B  ->  y  e.  dom  A ) )
4 vex 2577 . . . . . . . . . . 11  |-  y  e. 
_V
54elrn 4605 . . . . . . . . . 10  |-  ( y  e.  ran  B  <->  E. x  x B y )
64eldm 4560 . . . . . . . . . 10  |-  ( y  e.  dom  A  <->  E. z 
y A z )
75, 6imbi12i 232 . . . . . . . . 9  |-  ( ( y  e.  ran  B  ->  y  e.  dom  A
)  <->  ( E. x  x B y  ->  E. z 
y A z ) )
8 19.8a 1498 . . . . . . . . . . 11  |-  ( x B y  ->  E. x  x B y )
98imim1i 58 . . . . . . . . . 10  |-  ( ( E. x  x B y  ->  E. z 
y A z )  ->  ( x B y  ->  E. z 
y A z ) )
10 pm3.2 130 . . . . . . . . . . 11  |-  ( x B y  ->  (
y A z  -> 
( x B y  /\  y A z ) ) )
1110eximdv 1776 . . . . . . . . . 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 118 . . . . . . . 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 1776 . . . . . 6  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. y E. z ( x B y  /\  y A z ) ) )
16 excom 1570 . . . . . 6  |-  ( E. z E. y ( x B y  /\  y A z )  <->  E. y E. z ( x B y  /\  y A z ) )
1715, 16syl6ibr 155 . . . . 5  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. z E. y ( x B y  /\  y A z ) ) )
18 vex 2577 . . . . . . 7  |-  x  e. 
_V
19 vex 2577 . . . . . . 7  |-  z  e. 
_V
2018, 19opelco 4535 . . . . . 6  |-  ( <.
x ,  z >.  e.  ( A  o.  B
)  <->  E. y ( x B y  /\  y A z ) )
2120exbii 1512 . . . . 5  |-  ( E. z <. x ,  z
>.  e.  ( A  o.  B )  <->  E. z E. y ( x B y  /\  y A z ) )
2217, 21syl6ibr 155 . . . 4  |-  ( ran 
B  C_  dom  A  -> 
( E. y  x B y  ->  E. z <. x ,  z >.  e.  ( A  o.  B
) ) )
2318eldm 4560 . . . 4  |-  ( x  e.  dom  B  <->  E. y  x B y )
2418eldm2 4561 . . . 4  |-  ( x  e.  dom  ( A  o.  B )  <->  E. z <. x ,  z >.  e.  ( A  o.  B
) )
2522, 23, 243imtr4g 198 . . 3  |-  ( ran 
B  C_  dom  A  -> 
( x  e.  dom  B  ->  x  e.  dom  ( A  o.  B
) ) )
2625ssrdv 2979 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  B  C_  dom  ( A  o.  B ) )
272, 26eqssd 2990 1  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409    C_ wss 2945   <.cop 3406   class class class wbr 3792   dom cdm 4373   ran crn 4374    o. ccom 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384
This theorem is referenced by:  dmcoeq  4632  fnco  5035
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