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Theorem fnovrn 6243
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 4876 . . 3  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 df-ov 6099 . . . 4  |-  ( C F D )  =  ( F `  <. C ,  D >. )
3 fnfvelrn 5845 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( F `  <. C ,  D >. )  e.  ran  F )
42, 3syl5eqel 2527 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( C F D )  e. 
ran  F )
51, 4sylan2 474 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  ( C F D )  e. 
ran  F )
653impb 1183 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( C F D )  e.  ran  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756   <.cop 3888    X. cxp 4843   ran crn 4846    Fn wfn 5418   ` cfv 5423  (class class class)co 6096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431  df-ov 6099
This theorem is referenced by:  unirnioo  11394  ioorebas  11396  yonffthlem  15097  gsumval2a  15517  efginvrel2  16229  efgredleme  16245  efgcpbllemb  16257  mplsubrglem  17522  mplsubrglemOLD  17523  lecldbas  18828  blelrnps  19996  blelrn  19997  blssioo  20377  tgioo  20378  opnmbllem  21086  mbfdm  21111  mbfima  21115  isgrpo2  23689  tpr2rico  26347  dya2icoseg  26697  opnmbllem0  28432
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