Theorem f1oiOLD 6867

Description: Obsolete version of f1oi 6866 as of 10-Feb-2026. (Contributed by NM, 30-Apr-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
f1oiOLD	⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

Proof of Theorem f1oiOLD

Step	Hyp	Ref	Expression
1		fnresi 6671	. 2 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		cnvresid 6622	. . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
3	2	fneq1i 6639	. . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
4	1, 3	mpbir 234	. 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴
5		dff1o4 6836	. 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴))
6	1, 4, 5	mpbir2an 724	1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

This proof depends on syntax axioms:   I cid 5560  ^◡ccnv 5665   ↾ cres 5668   Fn wfn 6538  –1-1-onto→wf1o 6542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550

This theorem is used by: (None)