Theorem fnresin1 5473

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

Step	Hyp	Ref	Expression
1		inss1 3441	. 2 |- ( A i^i B ) C_ A
2		fnssres 5471	. 2 |- ( ( F Fn A /\ ( A i^i B ) C_ A ) -> ( F |` ( A i^i B ) ) Fn ( A i^i B ) )
3	1, 2	mpan2 425	1 |- ( F Fn A -> ( F |` ( A i^i B ) ) Fn ( A i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3210   C_ wss 3211   |` cres 4751   Fn wfn 5347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-fun 5354  df-fn 5355

This theorem is referenced by:  frecfnom  6632