Theorem fnresiOLD 6477

Description: Obsolete proof of fnresi 6476 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 6387	. . 3 ⊢ Fun I
2		funres 6397	. . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ Fun ( I ↾ 𝐴)
4		dmresi 5921	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
5		df-fn 6358	. 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
6	3, 4, 5	mpbir2an 709	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:   = wceq 1537   I cid 5459  dom cdm 5555   ↾ cres 5557  Fun wfun 6349   Fn wfn 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-res 5567  df-fun 6357  df-fn 6358

This theorem is referenced by: (None)