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Theorem fssres 6108
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 5930 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fnssres 6042 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
3 resss 5457 . . . . . . 7 (𝐹𝐶) ⊆ 𝐹
4 rnss 5386 . . . . . . 7 ((𝐹𝐶) ⊆ 𝐹 → ran (𝐹𝐶) ⊆ ran 𝐹)
53, 4ax-mp 5 . . . . . 6 ran (𝐹𝐶) ⊆ ran 𝐹
6 sstr 3644 . . . . . 6 ((ran (𝐹𝐶) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐶) ⊆ 𝐵)
75, 6mpan 706 . . . . 5 (ran 𝐹𝐵 → ran (𝐹𝐶) ⊆ 𝐵)
82, 7anim12i 589 . . . 4 (((𝐹 Fn 𝐴𝐶𝐴) ∧ ran 𝐹𝐵) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
98an32s 863 . . 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 5930 . 2 ((𝐹𝐶):𝐶𝐵 ↔ ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
1210, 11sylibr 224 1 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wss 3607  ran crn 5144  cres 5145   Fn wfn 5921  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  fssresd  6109  fssres2  6110  fresin  6111  fresaun  6113  f1ssres  6146  f2ndf  7328  elmapssres  7924  pmresg  7927  ralxpmap  7949  mapunen  8170  fofinf1o  8282  fseqenlem1  8885  inar1  9635  gruima  9662  addnqf  9808  mulnqf  9809  fseq1p1m1  12452  injresinj  12629  seqf1olem2  12881  wrdred1  13382  rlimres  14333  lo1res  14334  vdwnnlem1  15746  fsets  15938  resmhm  17406  resghm  17723  gsumzres  18356  gsumzadd  18368  gsum2dlem2  18416  dpjidcl  18503  ablfac1eu  18518  abvres  18887  znf1o  19948  islindf4  20225  kgencn  21407  ptrescn  21490  hmeores  21622  tsmsres  21994  tsmsmhm  21996  tsmsadd  21997  xrge0gsumle  22683  xrge0tsms  22684  ovolicc2lem4  23334  limcdif  23685  limcflf  23690  limcmo  23691  dvres  23720  dvres3a  23723  aannenlem1  24128  logcn  24438  dvlog  24442  dvlog2  24444  logtayl  24451  dvatan  24707  atancn  24708  efrlim  24741  amgm  24762  dchrelbas2  25007  redwlklem  26624  pthdivtx  26681  hhssabloilem  28246  hhssnv  28249  xrge0tsmsd  29913  cntmeas  30417  eulerpartlemt  30561  eulerpartlemmf  30565  eulerpartlemgvv  30566  subiwrd  30575  sseqp1  30585  wrdres  30745  poimirlem4  33543  mbfresfi  33586  mbfposadd  33587  itg2gt0cn  33595  sdclem2  33668  mzpcompact2lem  37631  eldiophb  37637  eldioph2  37642  cncfiooicclem1  40424  fouriersw  40766  sge0tsms  40915  psmeasure  41006  sssmf  41268  resmgmhm  42123  lindslinindimp2lem2  42573
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