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Theorem aoprssdm 28044
Description: Domain of closure of an operation. In contrast to oprssdm 6230, 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 4985 . 2  |-  Rel  ( S  X.  S )
2 opelxp 4910 . . 3  |-  ( <.
x ,  y >.  e.  ( S  X.  S
)  <->  ( x  e.  S  /\  y  e.  S ) )
3 df-aov 27954 . . . . 5  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
4 aoprssdm.1 . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )
53, 4syl5eqelr 2523 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( F''' <. x ,  y
>. )  e.  S
)
6 afvvdm 27983 . . . 4  |-  ( ( F''' <. x ,  y
>. )  e.  S  -> 
<. x ,  y >.  e.  dom  F )
75, 6syl 16 . . 3  |-  ( ( x  e.  S  /\  y  e.  S )  -> 
<. x ,  y >.  e.  dom  F )
82, 7sylbi 189 . 2  |-  ( <.
x ,  y >.  e.  ( S  X.  S
)  ->  <. x ,  y >.  e.  dom  F )
91, 8relssi 4969 1  |-  ( S  X.  S )  C_  dom  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    C_ wss 3322   <.cop 3819    X. cxp 4878   dom cdm 4880  '''cafv 27950   ((caov 27951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887  df-fv 5464  df-dfat 27952  df-afv 27953  df-aov 27954
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