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Theorem fovcl 5637
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
Hypothesis
Ref Expression
fovcl.1  |-  F :
( R  X.  S
) --> C
Assertion
Ref Expression
fovcl  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )

Proof of Theorem fovcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fovcl.1 . . 3  |-  F :
( R  X.  S
) --> C
2 ffnov 5636 . . . 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 269 . . 3  |-  ( F : ( R  X.  S ) --> C  ->  A. x  e.  R  A. y  e.  S  ( x F y )  e.  C )
41, 3ax-mp 7 . 2  |-  A. x  e.  R  A. y  e.  S  ( x F y )  e.  C
5 oveq1 5550 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
65eleq1d 2148 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  C  <->  ( A F y )  e.  C ) )
7 oveq2 5551 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eleq1d 2148 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  C  <->  ( A F B )  e.  C
) )
96, 8rspc2v 2714 . 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 ) )
104, 9mpi 15 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349    X. cxp 4369    Fn wfn 4927   -->wf 4928  (class class class)co 5543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-ov 5546
This theorem is referenced by:  ixxssxr  8999  fzof  9231  elfzoelz  9234  fzoval  9235
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