Theorem fnresi 5413

Description: Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)

Assertion

Ref	Expression
fnresi	|- ( _I |` A ) Fn A

Proof of Theorem fnresi

Step	Hyp	Ref	Expression
1		funi 5322	. . 3 |- Fun _I
2		funres 5331	. . 3 |- ( Fun _I -> Fun ( _I |` A ) )
3	1, 2	ax-mp 5	. 2 |- Fun ( _I |` A )
4		dmresi 5033	. 2 |- dom ( _I |` A ) = A
5		df-fn 5293	. 2 |- ( ( _I |` A ) Fn A <-> ( Fun ( _I |` A ) /\ dom ( _I |` A ) = A ) )
6	3, 4, 5	mpbir2an 945	1 |- ( _I |` A ) Fn A

Syntax hints:   = wceq 1373   _I cid 4353   dom cdm 4693   |` cres 4695   Fun wfun 5284   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-eu 2058  df-mo 2059  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-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-res 4705  df-fun 5292  df-fn 5293

This theorem is referenced by:  f1oi  5583  iordsmo  6406  omp1eomlem  7222  ctm  7237  xnn0nnen  10619