Theorem fnresdm 5240

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 5229	. 2 |- ( F Fn A -> Rel F )
2		fndm 5230	. . 3 |- ( F Fn A -> dom F = A )
3		eqimss 3156	. . 3 |- ( dom F = A -> dom F C_ A )
4	2, 3	syl 14	. 2 |- ( F Fn A -> dom F C_ A )
5		relssres 4865	. 2 |- ( ( Rel F /\ dom F C_ A ) -> ( F |` A ) = F )
6	1, 4, 5	syl2anc 409	1 |- ( F Fn A -> ( F |` A ) = F )

Syntax hints:   -> wi 4   = wceq 1332   C_ wss 3076   dom cdm 4547   |` cres 4549   Rel wrel 4552   Fn wfn 5126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-dm 4557  df-res 4559  df-fun 5133  df-fn 5134

This theorem is referenced by:  fnima  5249  fresin  5309  resasplitss  5310  fnsnsplitss  5627  fsnunfv  5629  fsnunres  5630  fnsnsplitdc  6409  fnfi  6833  fseq1p1m1  9905  facnn  10505  fac0  10506  dfrelog  12989