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Theorem faovcl 31707
Description: Closure law for an operation, analogous to fovcl 6382. (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 31706 . . . 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 2461 . . . . 5  |-  ( x  =  A  ->  F  =  F )
6 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
7 eqidd 2461 . . . . 5  |-  ( x  =  A  ->  y  =  y )
85, 6, 7aoveq123d 31685 . . . 4  |-  ( x  =  A  -> (( x F y))  = (( A F y))  )
98eleq1d 2529 . . 3  |-  ( x  =  A  ->  ( (( x F y))  e.  C  <-> (( A F y))  e.  C
) )
10 eqidd 2461 . . . . 5  |-  ( y  =  B  ->  F  =  F )
11 eqidd 2461 . . . . 5  |-  ( y  =  B  ->  A  =  A )
12 id 22 . . . . 5  |-  ( y  =  B  ->  y  =  B )
1310, 11, 12aoveq123d 31685 . . . 4  |-  ( y  =  B  -> (( A F y))  = (( A F B))  )
1413eleq1d 2529 . . 3  |-  ( y  =  B  ->  ( (( A F y))  e.  C  <-> (( A F B))  e.  C
) )
159, 14rspc2v 3216 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807    X. cxp 4990    Fn wfn 5574   -->wf 5575   ((caov 31622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-dfat 31623  df-afv 31624  df-aov 31625
This theorem is referenced by: (None)
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