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Theorem fnssresb 5630
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 5528 . 2  |-  ( ( F  |`  B )  Fn  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) )
2 fnfun 5615 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
3 funres 5564 . . . . 5  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
42, 3syl 17 . . . 4  |-  ( F  Fn  A  ->  Fun  ( F  |`  B ) )
54biantrurd 506 . . 3  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  B  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  B ) ) )
6 ssdmres 5236 . . . 4  |-  ( B 
C_  dom  F  <->  dom  ( F  |`  B )  =  B )
7 fndm 5617 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87sseq2d 3469 . . . 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 367    = wceq 1405    C_ wss 3413   dom cdm 4942    |` cres 4944   Fun wfun 5519    Fn wfn 5520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-res 4954  df-fun 5527  df-fn 5528
This theorem is referenced by:  fnssres  5631  plyreres  22863  redwlklem  24905  xrge0pluscn  28255  icoreresf  31257  fnbrafvb  37589  rhmsscrnghm  38325  rngcrescrhm  38384  rngcrescrhmALTV  38403
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