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Theorem aoprssdm 32045
Description: Domain of closure of an operation. In contrast to oprssdm 6450, 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 5115 . 2  |-  Rel  ( S  X.  S )
2 opelxp 5034 . . 3  |-  ( <.
x ,  y >.  e.  ( S  X.  S
)  <->  ( x  e.  S  /\  y  e.  S ) )
3 df-aov 31961 . . . . 5  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
4 aoprssdm.1 . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )
53, 4syl5eqelr 2560 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( F''' <. x ,  y
>. )  e.  S
)
6 afvvdm 31984 . . . 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 195 . 2  |-  ( <.
x ,  y >.  e.  ( S  X.  S
)  ->  <. x ,  y >.  e.  dom  F )
91, 8relssi 5099 1  |-  ( S  X.  S )  C_  dom  F
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    C_ wss 3481   <.cop 4038    X. cxp 5002   dom cdm 5004  '''cafv 31957   ((caov 31958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4511  df-xp 5010  df-rel 5011  df-fv 5601  df-dfat 31959  df-afv 31960  df-aov 31961
This theorem is referenced by: (None)
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