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Theorem fssres 5983
 Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fssres ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Proof of Theorem fssres
StepHypRef Expression
1 df-f 5808 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fnssres 5918 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
3 resss 5342 . . . . . . 7 (𝐹𝐶) ⊆ 𝐹
4 rnss 5275 . . . . . . 7 ((𝐹𝐶) ⊆ 𝐹 → ran (𝐹𝐶) ⊆ ran 𝐹)
53, 4ax-mp 5 . . . . . 6 ran (𝐹𝐶) ⊆ ran 𝐹
6 sstr 3576 . . . . . 6 ((ran (𝐹𝐶) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐶) ⊆ 𝐵)
75, 6mpan 702 . . . . 5 (ran 𝐹𝐵 → ran (𝐹𝐶) ⊆ 𝐵)
82, 7anim12i 588 . . . 4 (((𝐹 Fn 𝐴𝐶𝐴) ∧ ran 𝐹𝐵) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
98an32s 842 . . 3 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ 𝐶𝐴) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
101, 9sylanb 488 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
11 df-f 5808 . 2 ((𝐹𝐶):𝐶𝐵 ↔ ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
1210, 11sylibr 223 1 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ⊆ wss 3540  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  fssresd  5984  fssres2  5985  fresin  5986  fresaun  5988  f1ssres  6021  f2ndf  7170  elmapssres  7768  pmresg  7771  ralxpmap  7793  mapunen  8014  fofinf1o  8126  fseqenlem1  8730  inar1  9476  gruima  9503  addnqf  9649  mulnqf  9650  fseq1p1m1  12283  injresinj  12451  seqf1olem2  12703  rlimres  14137  lo1res  14138  vdwnnlem1  15537  fsets  15723  resmhm  17182  resghm  17499  gsumzres  18133  gsumzadd  18145  gsum2dlem2  18193  dpjidcl  18280  ablfac1eu  18295  abvres  18662  znf1o  19719  islindf4  19996  kgencn  21169  ptrescn  21252  hmeores  21384  tsmsres  21757  tsmsmhm  21759  tsmsadd  21760  xrge0gsumle  22444  xrge0tsms  22445  ovolicc2lem4  23095  limcdif  23446  limcflf  23451  limcmo  23452  dvres  23481  dvres3a  23484  aannenlem1  23887  logcn  24193  dvlog  24197  dvlog2  24199  logtayl  24206  dvatan  24462  atancn  24463  efrlim  24496  amgm  24517  dchrelbas2  24762  redwlk  26136  hhssabloilem  27502  hhssnv  27505  xrge0tsmsd  29116  cntmeas  29616  eulerpartlemt  29760  eulerpartlemmf  29764  eulerpartlemgvv  29765  subiwrd  29774  sseqp1  29784  wrdres  29943  poimirlem4  32583  mbfresfi  32626  mbfposadd  32627  itg2gt0cn  32635  sdclem2  32708  mzpcompact2lem  36332  eldiophb  36338  eldioph2  36343  cncfiooicclem1  38779  fouriersw  39124  sge0tsms  39273  psmeasure  39364  sssmf  39625  wrdred1  40240  red1wlklem  40880  pthdivtx  40935  resmgmhm  41588  lindslinindimp2lem2  42042
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