Theorem f1oiOLD 6820

Description: Obsolete version of f1oi 6819 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 6628	. 2 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		cnvresid 6578	. . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
3	2	fneq1i 6596	. . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
4	1, 3	mpbir 231	. 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴
5		dff1o4 6789	. 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴))
6	1, 4, 5	mpbir2an 712	1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

Syntax hints:   I cid 5525  ^◡ccnv 5630   ↾ cres 5633   Fn wfn 6494  –1-1-onto→wf1o 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506

This theorem is referenced by: (None)