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Theorem fnssres 5331
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
fnssres  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )

Proof of Theorem fnssres
StepHypRef Expression
1 fnssresb 5330 . 2  |-  ( F  Fn  A  ->  (
( F  |`  B )  Fn  B  <->  B  C_  A
) )
21biimpar 297 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    C_ wss 3131    |` cres 4630    Fn wfn 5213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-fun 5220  df-fn 5221
This theorem is referenced by:  fnresin1  5332  fnresin2  5333  fssres  5393  fvreseq  5621  fnreseql  5628  ffvresb  5681  fnressn  5704  ofres  6099  tfrlem1  6311  frecrdg  6411  resixp  6735  resfnfinfinss  6941  suplocexprlemell  7714  seq3feq2  10472  reeff1  11710  mgpf  13199  upxp  13811  uptx  13813  cnmpt1st  13827  cnmpt2nd  13828  ioocosf1o  14314
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