Theorem fnresdm 5495

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 5482	. 2 |- ( F Fn A -> Rel F )
2		fndm 5483	. . 3 |- ( F Fn A -> dom F = A )
3		eqimss 3302	. . 3 |- ( dom F = A -> dom F C_ A )
4	2, 3	syl 14	. 2 |- ( F Fn A -> dom F C_ A )
5		relssres 5103	. 2 |- ( ( Rel F /\ dom F C_ A ) -> ( F |` A ) = F )
6	1, 4, 5	syl2anc 415	1 |- ( F Fn A -> ( F |` A ) = F )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220   dom cdm 4776   |` cres 4778   Rel wrel 4781   Fn wfn 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-dm 4786  df-res 4788  df-fun 5382  df-fn 5383

This theorem is used by:  fnima  5505  fresin  5571  resasplitss  5572  fresaunres2disj  5573  fnsnsplitss  5917  fsnunfv  5919  fsnunres  5920  fnsnsplitdc  6782  mapunen  7155  fnfi  7254  fseq1p1m1  10518  facnn  11187  fac0  11188  gsump1  14247  gsumclfi  14249  gsumsubmclfi  14253  rnrhmsubrg  14669  cnfldms  15762  dfrelog  16089  eupthvdres  16921  domomsubct  17236