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Theorem fnssresb 5372
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
Assertion
Ref Expression
fnssresb  |-  ( F  Fn  A  ->  (
( F  |`  B )  Fn  B  <->  B  C_  A
) )

Proof of Theorem fnssresb
StepHypRef Expression
1 df-fn 5274 . 2  |-  ( ( F  |`  B )  Fn  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) )
2 fnfun 5357 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
3 funres 5309 . . . . 5  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
42, 3syl 15 . . . 4  |-  ( F  Fn  A  ->  Fun  ( F  |`  B ) )
54biantrurd 494 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) ) )
6 ssdmres 4993 . . . 4  |-  ( B 
C_  dom  F  <->  dom  ( F  |`  B )  =  B )
7 fndm 5359 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87sseq2d 3219 . . . 4  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
96, 8syl5bbr 250 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  B  C_  A
) )
105, 9bitr3d 246 . 2  |-  ( F  Fn  A  ->  (
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B )  <->  B  C_  A
) )
111, 10syl5bb 248 1  |-  ( F  Fn  A  ->  (
( F  |`  B )  Fn  B  <->  B  C_  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    C_ wss 3165   dom cdm 4705    |` cres 4707   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  fnssres  5373  plyreres  19679  xrge0pluscn  23337  prl2  25272  fnbrafvb  28122  redwlklem  28363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-fun 5273  df-fn 5274
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