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Theorem fnovrn 5995
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 4721 . . 3  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 df-ov 5861 . . . 4  |-  ( C F D )  =  ( F `  <. C ,  D >. )
3 fnfvelrn 5662 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( F `  <. C ,  D >. )  e.  ran  F )
42, 3syl5eqel 2367 . . 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 1684   <.cop 3643    X. cxp 4687   ran crn 4690    Fn wfn 5250   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  unirnioo  10743  ioorebas  10745  yonffthlem  14056  gsumval2a  14459  efginvrel2  15036  efgredleme  15052  efgcpbllemb  15064  mplsubrglem  16183  lecldbas  16949  blelrn  17967  blssioo  18301  tgioo  18302  opnmbllem  18956  mbfdm  18983  mbfima  18987  isgrpo2  20864  tpr2rico  23296  intrn  25599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861
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