Theorem fnresdm 5469

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 5456	. 2 |- ( F Fn A -> Rel F )
2		fndm 5457	. . 3 |- ( F Fn A -> dom F = A )
3		eqimss 3294	. . 3 |- ( dom F = A -> dom F C_ A )
4	2, 3	syl 14	. 2 |- ( F Fn A -> dom F C_ A )
5		relssres 5078	. 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 3213   dom cdm 4751   |` cres 4753   Rel wrel 4756   Fn wfn 5349

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 4230  ax-pow 4289  ax-pr 4324

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 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-dm 4761  df-res 4763  df-fun 5356  df-fn 5357

This theorem is referenced by:  fnima  5479  fresin  5545  resasplitss  5546  fresaunres2disj  5547  fnsnsplitss  5885  fsnunfv  5887  fsnunres  5888  fnsnsplitdc  6740  mapunen  7106  fnfi  7205  fseq1p1m1  10432  facnn  11093  fac0  11094  rnrhmsubrg  14414  cnfldms  15418  dfrelog  15742  eupthvdres  16487  domomsubct  16792  gfsump1  16885  gfsumcl  16887