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Theorem fnresin2 5562
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
fnresin2  |-  ( F  Fn  A  ->  ( F  |`  ( B  i^i  A ) )  Fn  ( B  i^i  A ) )

Proof of Theorem fnresin2
StepHypRef Expression
1 inss2 3564 . 2  |-  ( B  i^i  A )  C_  A
2 fnssres 5560 . 2  |-  ( ( F  Fn  A  /\  ( B  i^i  A ) 
C_  A )  -> 
( F  |`  ( B  i^i  A ) )  Fn  ( B  i^i  A ) )
31, 2mpan2 654 1  |-  ( F  Fn  A  ->  ( F  |`  ( B  i^i  A ) )  Fn  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 3321    C_ wss 3322    |` cres 4882    Fn wfn 5451
This theorem is referenced by:  tfrlem5  6643  hashresfn  24158  resfnfinfin  28088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-fun 5458  df-fn 5459
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