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Theorem fnssresb 5698
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 5596 . 2  |-  ( ( F  |`  B )  Fn  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) )
2 fnfun 5683 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
3 funres 5632 . . . . 5  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
42, 3syl 17 . . . 4  |-  ( F  Fn  A  ->  Fun  ( F  |`  B ) )
54biantrurd 510 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) ) )
6 ssdmres 5138 . . . 4  |-  ( B 
C_  dom  F  <->  dom  ( F  |`  B )  =  B )
7 fndm 5685 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87sseq2d 3489 . . . 4  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
96, 8syl5bbr 262 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  B  C_  A
) )
105, 9bitr3d 258 . 2  |-  ( F  Fn  A  ->  (
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B )  <->  B  C_  A
) )
111, 10syl5bb 260 1  |-  ( F  Fn  A  ->  (
( F  |`  B )  Fn  B  <->  B  C_  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    C_ wss 3433   dom cdm 4846    |` cres 4848   Fun wfun 5587    Fn wfn 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-res 4858  df-fun 5595  df-fn 5596
This theorem is referenced by:  fnssres  5699  plyreres  23213  redwlklem  25311  xrge0pluscn  28735  icoreresf  31690  fnbrafvb  38276  rhmsscrnghm  39090  rngcrescrhm  39149  rngcrescrhmALTV  39168
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