Theorem fnresiOLD 6558

Description: Obsolete proof of fnresi 6557 as of 27-Dec-2023. (Contributed by NM, 27-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fnresiOLD	⊢ ( I ↾ 𝐴) Fn 𝐴

Proof of Theorem fnresiOLD

Step	Hyp	Ref	Expression
1		funi 6462	. . 3 ⊢ Fun I
2		funres 6472	. . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ Fun ( I ↾ 𝐴)
4		dmresi 5958	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
5		df-fn 6433	. 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
6	3, 4, 5	mpbir2an 707	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:   = wceq 1541   I cid 5487  dom cdm 5588   ↾ cres 5590  Fun wfun 6424   Fn wfn 6425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-res 5600  df-fun 6432  df-fn 6433

This theorem is referenced by: (None)