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Theorem faovcl 28168
Description: Closure law for an operation, analogous to fovcl 5965. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
faovcl.1  |-  F :
( R  X.  S
) --> C
Assertion
Ref Expression
faovcl  |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )

Proof of Theorem faovcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 faovcl.1 . . 3  |-  F :
( R  X.  S
) --> C
2 ffnaov 28167 . . . 4  |-  ( F : ( R  X.  S ) --> C  <->  ( F  Fn  ( R  X.  S
)  /\  A. x  e.  R  A. y  e.  S (( x F
y))  e.  C ) )
32simprbi 450 . . 3  |-  ( F : ( R  X.  S ) --> C  ->  A. x  e.  R  A. y  e.  S (( x F y))  e.  C
)
41, 3ax-mp 8 . 2  |-  A. x  e.  R  A. y  e.  S (( x F
y))  e.  C
5 eqidd 2297 . . . . 5  |-  ( x  =  A  ->  F  =  F )
6 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
7 eqidd 2297 . . . . 5  |-  ( x  =  A  ->  y  =  y )
85, 6, 7aoveq123d 28146 . . . 4  |-  ( x  =  A  -> (( x F y))  = (( A F y))  )
98eleq1d 2362 . . 3  |-  ( x  =  A  ->  ( (( x F y))  e.  C  <-> (( A F y))  e.  C
) )
10 eqidd 2297 . . . . 5  |-  ( y  =  B  ->  F  =  F )
11 eqidd 2297 . . . . 5  |-  ( y  =  B  ->  A  =  A )
12 id 19 . . . . 5  |-  ( y  =  B  ->  y  =  B )
1310, 11, 12aoveq123d 28146 . . . 4  |-  ( y  =  B  -> (( A F y))  = (( A F B))  )
1413eleq1d 2362 . . 3  |-  ( y  =  B  ->  ( (( A F y))  e.  C  <-> (( A F B))  e.  C
) )
159, 14rspc2v 2903 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A. x  e.  R  A. y  e.  S (( x F y))  e.  C  -> (( A F B))  e.  C ) )
164, 15mpi 16 1  |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    X. cxp 4703    Fn wfn 5266   -->wf 5267   ((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-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-eu 2160  df-mo 2161  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-sbc 3005  df-csb 3095  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-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-dfat 28077  df-afv 28078  df-aov 28079
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