Theorem fnresi 5501

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 5409	. . 3 |- Fun _I
2		funres 5418	. . 3 |- ( Fun _I -> Fun ( _I |` A ) )
3	1, 2	ax-mp 5	. 2 |- Fun ( _I |` A )
4		dmresi 5118	. 2 |- dom ( _I |` A ) = A
5		df-fn 5380	. 2 |- ( ( _I |` A ) Fn A <-> ( Fun ( _I |` A ) /\ dom ( _I |` A ) = A ) )
6	3, 4, 5	mpbir2an 955	1 |- ( _I |` A ) Fn A

This proof depends on syntax axioms:   = wceq 1402   _I cid 4433   dom cdm 4774   |` cres 4776   Fun wfun 5371   Fn wfn 5372

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  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 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-fun 5379  df-fn 5380

This theorem is used by:  f1oi  5679  iordsmo  6568  omp1eomlem  7434  ctm  7449  xnn0nnen  10874