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Theorem fnssres 5329
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 5328 . 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 3129  cres 4628   Fn wfn 5211
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-fun 5218  df-fn 5219
This theorem is referenced by:  fnresin1  5330  fnresin2  5331  fssres  5391  fvreseq  5619  fnreseql  5626  ffvresb  5679  fnressn  5702  ofres  6096  tfrlem1  6308  frecrdg  6408  resixp  6732  resfnfinfinss  6938  suplocexprlemell  7711  seq3feq2  10469  reeff1  11707  mgpf  13192  upxp  13742  uptx  13744  cnmpt1st  13758  cnmpt2nd  13759  ioocosf1o  14245
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