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Theorem fovcl 5842
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 5841 . . . 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 271 . . 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 oveq1 5747 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
65eleq1d 2184 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  C  <->  ( A F y )  e.  C ) )
7 oveq2 5748 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eleq1d 2184 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  C  <->  ( A F B )  e.  C
) )
96, 8rspc2v 2774 . 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 103    = wceq 1314    e. wcel 1463   A.wral 2391    X. cxp 4505    Fn wfn 5086   -->wf 5087  (class class class)co 5740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743
This theorem is referenced by:  xaddcl  9594  ixxssxr  9634  fzof  9872  elfzoelz  9875  fzoval  9876  addcncntoplem  12626
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