Theorem fnresin1 5322

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 3353	. 2 |- ( A i^i B ) C_ A
2		fnssres 5321	. 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 3126   C_ wss 3127   |` cres 4622   Fn wfn 5203

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-res 4632  df-fun 5210  df-fn 5211

This theorem is referenced by:  frecfnom  6392