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Theorem fnresin1 5701
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
fnresin1  |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )

Proof of Theorem fnresin1
StepHypRef Expression
1 inss1 3714 . 2  |-  ( A  i^i  B )  C_  A
2 fnssres 5700 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  B ) 
C_  A )  -> 
( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
31, 2mpan2 671 1  |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    i^i cin 3470    C_ wss 3471    |` cres 5010    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-fun 5596  df-fn 5597
This theorem is referenced by:  fnresin  27612  wfrlem4  29520  frrlem4  29564
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