Theorem funres 6518

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 (𝐹 ↾ 𝐴))

Proof of Theorem funres

Step	Hyp	Ref	Expression
1		resss 5945	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		funss 6495	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3897   ↾ cres 5613  Fun wfun 6470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ss 3914  df-br 5087  df-opab 5149  df-rel 5618  df-cnv 5619  df-co 5620  df-res 5623  df-fun 6478

This theorem is referenced by:  funresd  6519  fores  6740  resfunexg  7144  funfvima  7159  funiunfv  7177  fprlem1  8225  smores  8267  smores2  8269  frfnom  8349  sbthlem7  9001  fsuppres  9272  ordtypelem4  9402  wdomima2g  9467  imadomg  10420  hashres  14340  hashimarn  14342  setsfun  17077  setsfun0  17078  lubfun  18251  glbfun  18264  qtoptop2  23609  volf  25452  nolesgn2ores  27606  nosupres  27641  nosupbnd2lem1  27649  noetasuplem4  27670  noetainflem4  27674  onsiso  28200  bdayn0sf1o  28290  uhgrspansubgrlem  29263  upgrres  29279  umgrres  29280  hlimf  31209  fsuppcurry1  32699  fsuppcurry2  32700  eulerpartlemmf  34380  eulerpartlemgvv  34381  bj-funidres  37185  imadomfi  42035  funcoressn  47073  fundmdfat  47160  afvelrn  47199  dmfcoafv  47206  aovmpt4g  47232  fundmafv2rnb  47261