Theorem fnresdm 5490

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 5477	. 2 |- ( F Fn A -> Rel F )
2		fndm 5478	. . 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 5099	. 2 |- ( ( Rel F /\ dom F C_ A ) -> ( F |` A ) = F )
6	1, 4, 5	syl2anc 415	1 |- ( F Fn A -> ( F |` A ) = F )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220   dom cdm 4772   |` cres 4774   Rel wrel 4777   Fn wfn 5370

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 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 4247  ax-pow 4309  ax-pr 4344

This theorem 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 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-dm 4782  df-res 4784  df-fun 5377  df-fn 5378

This theorem is referenced by:  fnima  5500  fresin  5566  resasplitss  5567  fresaunres2disj  5568  fnsnsplitss  5908  fsnunfv  5910  fsnunres  5911  fnsnsplitdc  6771  mapunen  7144  fnfi  7243  fseq1p1m1  10482  facnn  11146  fac0  11147  gsump1  14137  gsumclfi  14139  gsumsubmclfi  14143  rnrhmsubrg  14536  cnfldms  15563  dfrelog  15887  eupthvdres  16633  domomsubct  16948