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Theorem fnovrn 5886
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 4541 . . 3  |-  ( ( C  e.  A  /\  D  e.  B )  -> 
<. C ,  D >.  e.  ( A  X.  B
) )
2 df-ov 5745 . . . 4  |-  ( C F D )  =  ( F `  <. C ,  D >. )
3 fnfvelrn 5520 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( F `  <. C ,  D >. )  e.  ran  F )
42, 3eqeltrid 2204 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  <. C ,  D >.  e.  ( A  X.  B
) )  ->  ( C F D )  e. 
ran  F )
51, 4sylan2 284 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  ( C F D )  e. 
ran  F )
653impb 1162 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 103    /\ w3a 947    e. wcel 1465   <.cop 3500    X. cxp 4507   ran crn 4510    Fn wfn 5088   ` cfv 5093  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-ov 5745
This theorem is referenced by:  unirnioo  9724  ioorebasg  9726  blelrnps  12515  blelrn  12516  xmettx  12606
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