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Theorem fnovrn 6433
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 4857 . . 3  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 df-ov 6283 . . . 4  |-  ( C F D )  =  ( F `  <. C ,  D >. )
3 fnfvelrn 6008 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( F `  <. C ,  D >. )  e.  ran  F )
42, 3syl5eqel 2496 . . 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 1195 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 976    e. wcel 1844   <.cop 3980    X. cxp 4823   ran crn 4826    Fn wfn 5566   ` cfv 5571  (class class class)co 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-iota 5535  df-fun 5573  df-fn 5574  df-fv 5579  df-ov 6283
This theorem is referenced by:  unirnioo  11680  ioorebas  11682  yonffthlem  15877  gsumval2a  16232  efginvrel2  17071  efgredleme  17087  efgcpbllemb  17099  mplsubrglem  18422  mplsubrglemOLD  18423  lecldbas  20015  blelrnps  21213  blelrn  21214  blssioo  21594  tgioo  21595  opnmbllem  22304  mbfdm  22329  mbfima  22333  isgrpo2  25626  tpr2rico  28360  dya2icoseg  28738  opnmbllem0  31435
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