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Theorem fnssresb 5684
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 5582 . 2  |-  ( ( F  |`  B )  Fn  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) )
2 fnfun 5669 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
3 funres 5618 . . . . 5  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
42, 3syl 16 . . . 4  |-  ( F  Fn  A  ->  Fun  ( F  |`  B ) )
54biantrurd 508 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) ) )
6 ssdmres 5286 . . . 4  |-  ( B 
C_  dom  F  <->  dom  ( F  |`  B )  =  B )
7 fndm 5671 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87sseq2d 3525 . . . 4  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
96, 8syl5bbr 259 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  B  C_  A
) )
105, 9bitr3d 255 . 2  |-  ( F  Fn  A  ->  (
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B )  <->  B  C_  A
) )
111, 10syl5bb 257 1  |-  ( F  Fn  A  ->  (
( F  |`  B )  Fn  B  <->  B  C_  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    C_ wss 3469   dom cdm 4992    |` cres 4994   Fun wfun 5573    Fn wfn 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-res 5004  df-fun 5581  df-fn 5582
This theorem is referenced by:  fnssres  5685  plyreres  22406  redwlklem  24269  xrge0pluscn  27544  fnbrafvb  31661
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