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Theorem faovcl 28042
Description: Closure law for an operation, analogous to fovcl 6177. (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 28041 . . . 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 452 . . 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 2439 . . . . 5  |-  ( x  =  A  ->  F  =  F )
6 id 21 . . . . 5  |-  ( x  =  A  ->  x  =  A )
7 eqidd 2439 . . . . 5  |-  ( x  =  A  ->  y  =  y )
85, 6, 7aoveq123d 28020 . . . 4  |-  ( x  =  A  -> (( x F y))  = (( A F y))  )
98eleq1d 2504 . . 3  |-  ( x  =  A  ->  ( (( x F y))  e.  C  <-> (( A F y))  e.  C
) )
10 eqidd 2439 . . . . 5  |-  ( y  =  B  ->  F  =  F )
11 eqidd 2439 . . . . 5  |-  ( y  =  B  ->  A  =  A )
12 id 21 . . . . 5  |-  ( y  =  B  ->  y  =  B )
1310, 11, 12aoveq123d 28020 . . . 4  |-  ( y  =  B  -> (( A F y))  = (( A F B))  )
1413eleq1d 2504 . . 3  |-  ( y  =  B  ->  ( (( A F y))  e.  C  <-> (( A F B))  e.  C
) )
159, 14rspc2v 3060 . 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 17 1  |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    X. cxp 4878    Fn wfn 5451   -->wf 5452   ((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-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-eu 2287  df-mo 2288  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-sbc 3164  df-csb 3254  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-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-dfat 27952  df-afv 27953  df-aov 27954
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