Theorem fnresdm 5492

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 5479	. 2 |- ( F Fn A -> Rel F )
2		fndm 5480	. . 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 5101	. 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 4774   |` cres 4776   Rel wrel 4779   Fn wfn 5372

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 4249  ax-pow 4311  ax-pr 4346

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 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-res 4786  df-fun 5379  df-fn 5380

This theorem is used by:  fnima  5502  fresin  5568  resasplitss  5569  fresaunres2disj  5570  fnsnsplitss  5914  fsnunfv  5916  fsnunres  5917  fnsnsplitdc  6778  mapunen  7151  fnfi  7250  fseq1p1m1  10501  facnn  11165  fac0  11166  gsump1  14157  gsumclfi  14159  gsumsubmclfi  14163  rnrhmsubrg  14560  cnfldms  15637  dfrelog  15961  eupthvdres  16716  domomsubct  17031