Theorem fnresi 5315

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 5230	. . 3 |- Fun _I
2		funres 5239	. . 3 |- ( Fun _I -> Fun ( _I |` A ) )
3	1, 2	ax-mp 5	. 2 |- Fun ( _I |` A )
4		dmresi 4946	. 2 |- dom ( _I |` A ) = A
5		df-fn 5201	. 2 |- ( ( _I |` A ) Fn A <-> ( Fun ( _I |` A ) /\ dom ( _I |` A ) = A ) )
6	3, 4, 5	mpbir2an 937	1 |- ( _I |` A ) Fn A

Syntax hints:   = wceq 1348   _I cid 4273   dom cdm 4611   |` cres 4613   Fun wfun 5192   Fn wfn 5193

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-fun 5200  df-fn 5201

This theorem is referenced by:  f1oi  5480  iordsmo  6276  omp1eomlem  7071  ctm  7086