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Theorem fnovex 5563
Description: The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
Assertion
Ref Expression
fnovex  |-  ( ( F  Fn  ( C  X.  D )  /\  A  e.  C  /\  B  e.  D )  ->  ( A F B )  e.  _V )

Proof of Theorem fnovex
StepHypRef Expression
1 df-ov 5540 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 opelxp 4394 . . . 4  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
3 funfvex 5217 . . . . 5  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( F `  <. A ,  B >. )  e.  _V )
43funfni 5024 . . . 4  |-  ( ( F  Fn  ( C  X.  D )  /\  <. A ,  B >.  e.  ( C  X.  D
) )  ->  ( F `  <. A ,  B >. )  e.  _V )
52, 4sylan2br 282 . . 3  |-  ( ( F  Fn  ( C  X.  D )  /\  ( A  e.  C  /\  B  e.  D
) )  ->  ( F `  <. A ,  B >. )  e.  _V )
653impb 1135 . 2  |-  ( ( F  Fn  ( C  X.  D )  /\  A  e.  C  /\  B  e.  D )  ->  ( F `  <. A ,  B >. )  e.  _V )
71, 6syl5eqel 2166 1  |-  ( ( F  Fn  ( C  X.  D )  /\  A  e.  C  /\  B  e.  D )  ->  ( A F B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    e. wcel 1434   _Vcvv 2602   <.cop 3403    X. cxp 4363    Fn wfn 4921   ` cfv 4926  (class class class)co 5537
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 3898  ax-pow 3950  ax-pr 3966
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-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fn 4929  df-fv 4934  df-ov 5540
This theorem is referenced by:  ovelrn  5674  fnofval  5746  fzen  9127
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