MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnresin1 Unicode version

Theorem fnresin1 5500
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 3505 . 2  |-  ( A  i^i  B )  C_  A
2 fnssres 5499 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  B ) 
C_  A )  -> 
( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
31, 2mpan2 653 1  |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 3263    C_ wss 3264    |` cres 4821    Fn wfn 5390
This theorem is referenced by:  tfrlem5  6578  wfrlem4  25284  frrlem4  25309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-res 4831  df-fun 5397  df-fn 5398
  Copyright terms: Public domain W3C validator