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Theorem funres 4884
 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 4578 . 2 (𝐹A) ⊆ 𝐹
2 funss 4863 . 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 2911   ↾ cres 4290  Fun wfun 4839 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-res 4300  df-fun 4847 This theorem is referenced by:  fnssresb  4954  fnresi  4959  fores  5058  respreima  5238  resfunexg  5325  funfvima  5333  smores  5848  smores2  5850
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