Theorem f1oiOLD 6860

Description: Obsolete version of f1oi 6859 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 6664	. 2 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		cnvresid 6615	. . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
3	2	fneq1i 6632	. . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
4	1, 3	mpbir 234	. 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴
5		dff1o4 6829	. 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴))
6	1, 4, 5	mpbir2an 723	1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

Syntax hints:   I cid 5555  ^◡ccnv 5660   ↾ cres 5663   Fn wfn 6531  –1-1-onto→wf1o 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is referenced by: (None)