Theorem f1oiOLD 6810

Description: Obsolete version of f1oi 6809 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 6618	. 2 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		cnvresid 6568	. . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
3	2	fneq1i 6586	. . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
4	1, 3	mpbir 231	. 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴
5		dff1o4 6779	. 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴))
6	1, 4, 5	mpbir2an 711	1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

Syntax hints:   I cid 5515  ^◡ccnv 5620   ↾ cres 5623   Fn wfn 6484  –1-1-onto→wf1o 6488

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496

This theorem is referenced by: (None)