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Theorem faovcl 28060
Description: Closure law for an operation, analogous to fovcl 5949. (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 28059 . . . 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 2284 . . . . 5  |-  ( x  =  A  ->  F  =  F )
6 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
7 eqidd 2284 . . . . 5  |-  ( x  =  A  ->  y  =  y )
85, 6, 7aoveq123d 28038 . . . 4  |-  ( x  =  A  -> (( x F y))  = (( A F y))  )
98eleq1d 2349 . . 3  |-  ( x  =  A  ->  ( (( x F y))  e.  C  <-> (( A F y))  e.  C
) )
10 eqidd 2284 . . . . 5  |-  ( y  =  B  ->  F  =  F )
11 eqidd 2284 . . . . 5  |-  ( y  =  B  ->  A  =  A )
12 id 19 . . . . 5  |-  ( y  =  B  ->  y  =  B )
1310, 11, 12aoveq123d 28038 . . . 4  |-  ( y  =  B  -> (( A F y))  = (( A F B))  )
1413eleq1d 2349 . . 3  |-  ( y  =  B  ->  ( (( A F y))  e.  C  <-> (( A F B))  e.  C
) )
159, 14rspc2v 2890 . 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 1623    e. wcel 1684   A.wral 2543    X. cxp 4687    Fn wfn 5250   -->wf 5251   ((caov 27973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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