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Theorem fnovrn 6011
Description: An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
Assertion
Ref Expression
fnovrn  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( C F D )  e.  ran  F
)

Proof of Theorem fnovrn
StepHypRef Expression
1 opelxpi 4737 . . 3  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 df-ov 5877 . . . 4  |-  ( C F D )  =  ( F `  <. C ,  D >. )
3 fnfvelrn 5678 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( F `  <. C ,  D >. )  e.  ran  F )
42, 3syl5eqel 2380 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( C F D )  e. 
ran  F )
51, 4sylan2 460 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  ( C F D )  e. 
ran  F )
653impb 1147 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( C F D )  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696   <.cop 3656    X. cxp 4703   ran crn 4706    Fn wfn 5266   ` cfv 5271  (class class class)co 5874
This theorem is referenced by:  unirnioo  10759  ioorebas  10761  yonffthlem  14072  gsumval2a  14475  efginvrel2  15052  efgredleme  15068  efgcpbllemb  15080  mplsubrglem  16199  lecldbas  16965  blelrn  17983  blssioo  18317  tgioo  18318  opnmbllem  18972  mbfdm  18999  mbfima  19003  isgrpo2  20880  tpr2rico  23311  intrn  25702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877
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