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Theorem fnssresb 5521
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 5419 . 2  |-  ( ( F  |`  B )  Fn  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) )
2 fnfun 5506 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
3 funres 5455 . . . . 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 5130 . . . 4  |-  ( B 
C_  dom  F  <->  dom  ( F  |`  B )  =  B )
7 fndm 5508 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87sseq2d 3382 . . . 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 1369    C_ wss 3326   dom cdm 4838    |` cres 4840   Fun wfun 5410    Fn wfn 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-fun 5418  df-fn 5419
This theorem is referenced by:  fnssres  5522  plyreres  21747  redwlklem  23502  xrge0pluscn  26368  fnbrafvb  30057
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