Theorem fnresiOLD 6562

Description: Obsolete proof of fnresi 6561 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 6466	. . 3 ⊢ Fun I
2		funres 6476	. . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ Fun ( I ↾ 𝐴)
4		dmresi 5961	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
5		df-fn 6436	. 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
6	3, 4, 5	mpbir2an 708	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:   = wceq 1539   I cid 5488  dom cdm 5589   ↾ cres 5591  Fun wfun 6427   Fn wfn 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-fun 6435  df-fn 6436

This theorem is referenced by: (None)