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Theorem faovcl 37666
Description: Closure law for an operation, analogous to fovcl 6390. (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 37665 . . . 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 464 . . 3  |-  ( F : ( R  X.  S ) --> C  ->  A. x  e.  R  A. y  e.  S (( x F y))  e.  C
)
41, 3ax-mp 5 . 2  |-  A. x  e.  R  A. y  e.  S (( x F
y))  e.  C
5 eqidd 2405 . . . . 5  |-  ( x  =  A  ->  F  =  F )
6 id 23 . . . . 5  |-  ( x  =  A  ->  x  =  A )
7 eqidd 2405 . . . . 5  |-  ( x  =  A  ->  y  =  y )
85, 6, 7aoveq123d 37644 . . . 4  |-  ( x  =  A  -> (( x F y))  = (( A F y))  )
98eleq1d 2473 . . 3  |-  ( x  =  A  ->  ( (( x F y))  e.  C  <-> (( A F y))  e.  C
) )
10 eqidd 2405 . . . . 5  |-  ( y  =  B  ->  F  =  F )
11 eqidd 2405 . . . . 5  |-  ( y  =  B  ->  A  =  A )
12 id 23 . . . . 5  |-  ( y  =  B  ->  y  =  B )
1310, 11, 12aoveq123d 37644 . . . 4  |-  ( y  =  B  -> (( A F y))  = (( A F B))  )
1413eleq1d 2473 . . 3  |-  ( y  =  B  ->  ( (( A F y))  e.  C  <-> (( A F B))  e.  C
) )
159, 14rspc2v 3171 . 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 21 1  |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756    X. cxp 4823    Fn wfn 5566   -->wf 5567   ((caov 37581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-dfat 37582  df-afv 37583  df-aov 37584
This theorem is referenced by: (None)
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