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Theorem fnssres 5330
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
fnssres ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)

Proof of Theorem fnssres
StepHypRef Expression
1 fnssresb 5329 . 2 (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
21biimpar 297 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wss 3130  cres 4629   Fn wfn 5212
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-res 4639  df-fun 5219  df-fn 5220
This theorem is referenced by:  fnresin1  5331  fnresin2  5332  fssres  5392  fvreseq  5620  fnreseql  5627  ffvresb  5680  fnressn  5703  ofres  6097  tfrlem1  6309  frecrdg  6409  resixp  6733  resfnfinfinss  6939  suplocexprlemell  7712  seq3feq2  10470  reeff1  11708  mgpf  13194  upxp  13775  uptx  13777  cnmpt1st  13791  cnmpt2nd  13792  ioocosf1o  14278
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