Theorem fnresdm 5404

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 5391	. 2 |- ( F Fn A -> Rel F )
2		fndm 5392	. . 3 |- ( F Fn A -> dom F = A )
3		eqimss 3255	. . 3 |- ( dom F = A -> dom F C_ A )
4	2, 3	syl 14	. 2 |- ( F Fn A -> dom F C_ A )
5		relssres 5016	. 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 1373   C_ wss 3174   dom cdm 4693   |` cres 4695   Rel wrel 4698   Fn wfn 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-dm 4703  df-res 4705  df-fun 5292  df-fn 5293

This theorem is referenced by:  fnima  5414  fresin  5476  resasplitss  5477  fnsnsplitss  5806  fsnunfv  5808  fsnunres  5809  fnsnsplitdc  6614  fnfi  7064  fseq1p1m1  10251  facnn  10909  fac0  10910  rnrhmsubrg  14129  cnfldms  15123  dfrelog  15447  domomsubct  16140