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Theorem fnovex 5641
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 5618 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 opelxp 4442 . . . 4  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
3 funfvex 5287 . . . . 5  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( F `  <. A ,  B >. )  e.  _V )
43funfni 5081 . . . 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 1137 . 2  |-  ( ( F  Fn  ( C  X.  D )  /\  A  e.  C  /\  B  e.  D )  ->  ( F `  <. A ,  B >. )  e.  _V )
71, 6syl5eqel 2171 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 922    e. wcel 1436   _Vcvv 2615   <.cop 3434    X. cxp 4411    Fn wfn 4978   ` cfv 4983  (class class class)co 5615
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fn 4986  df-fv 4991  df-ov 5618
This theorem is referenced by:  ovelrn  5752  fnofval  5824  mapsnen  6482  map1  6483  mapen  6516  mapdom1g  6517  mapxpen  6518  xpmapenlem  6519  fzen  9392  hashfacen  10141
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