Theorem fnresiOLD 6546

Description: Obsolete proof of fnresi 6545 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 6450	. . 3 ⊢ Fun I
2		funres 6460	. . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ Fun ( I ↾ 𝐴)
4		dmresi 5950	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
5		df-fn 6421	. 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 1539   I cid 5479  dom cdm 5580   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-fun 6420  df-fn 6421

This theorem is referenced by: (None)