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Theorem aoprssdm 28170
Description: Domain of closure of an operation. In contrast to oprssdm 6018, no additional property for S (
-.  (/)  e.  S) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aoprssdm.1  |-  ( ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )
Assertion
Ref Expression
aoprssdm  |-  ( S  X.  S )  C_  dom  F
Distinct variable groups:    x, y, S    x, F, y

Proof of Theorem aoprssdm
StepHypRef Expression
1 relxp 4810 . 2  |-  Rel  ( S  X.  S )
2 opelxp 4735 . . 3  |-  ( <.
x ,  y >.  e.  ( S  X.  S
)  <->  ( x  e.  S  /\  y  e.  S ) )
3 df-aov 28079 . . . . 5  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
4 aoprssdm.1 . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )
53, 4syl5eqelr 2381 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( F''' <. x ,  y
>. )  e.  S
)
6 afvvdm 28109 . . . 4  |-  ( ( F''' <. x ,  y
>. )  e.  S  -> 
<. x ,  y >.  e.  dom  F )
75, 6syl 15 . . 3  |-  ( ( x  e.  S  /\  y  e.  S )  -> 
<. x ,  y >.  e.  dom  F )
82, 7sylbi 187 . 2  |-  ( <.
x ,  y >.  e.  ( S  X.  S
)  ->  <. x ,  y >.  e.  dom  F )
91, 8relssi 4794 1  |-  ( S  X.  S )  C_  dom  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    C_ wss 3165   <.cop 3656    X. cxp 4703   dom cdm 4705  '''cafv 28075   ((caov 28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712  df-fv 5279  df-dfat 28077  df-afv 28078  df-aov 28079
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