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Theorem funres 4969
 Description: A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
funres (Fun 𝐹 → Fun (𝐹𝐴))

Proof of Theorem funres
StepHypRef Expression
1 resss 4663 . 2 (𝐹𝐴) ⊆ 𝐹
2 funss 4948 . 2 ((𝐹𝐴) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐴)))
31, 2ax-mp 7 1 (Fun 𝐹 → Fun (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 2945   ↾ cres 4375  Fun wfun 4924 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-br 3793  df-opab 3847  df-rel 4380  df-cnv 4381  df-co 4382  df-res 4385  df-fun 4932 This theorem is referenced by:  fnssresb  5039  fnresi  5044  fores  5143  respreima  5323  resfunexg  5410  funfvima  5418  smores  5938  smores2  5940
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