Theorem fnresdm 5473

Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)

Assertion

Ref	Expression
fnresdm	|- ( F Fn A -> ( F |` A ) = F )

Proof of Theorem fnresdm

Step	Hyp	Ref	Expression
1		fnrel 5460	. 2 |- ( F Fn A -> Rel F )
2		fndm 5461	. . 3 |- ( F Fn A -> dom F = A )
3		eqimss 3296	. . 3 |- ( dom F = A -> dom F C_ A )
4	2, 3	syl 14	. 2 |- ( F Fn A -> dom F C_ A )
5		relssres 5082	. 2 |- ( ( Rel F /\ dom F C_ A ) -> ( F |` A ) = F )
6	1, 4, 5	syl2anc 411	1 |- ( F Fn A -> ( F |` A ) = F )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3214   dom cdm 4755   |` cres 4757   Rel wrel 4760   Fn wfn 5353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-xp 4761  df-rel 4762  df-dm 4765  df-res 4767  df-fun 5360  df-fn 5361

This theorem is referenced by:  fnima  5483  fresin  5549  resasplitss  5550  fresaunres2disj  5551  fnsnsplitss  5889  fsnunfv  5891  fsnunres  5892  fnsnsplitdc  6752  mapunen  7118  fnfi  7217  fseq1p1m1  10454  facnn  11118  fac0  11119  gfsump1  14113  gfsumcl  14115  rnrhmsubrg  14503  cnfldms  15532  dfrelog  15856  eupthvdres  16601  domomsubct  16916