Theorem cnvresid

Description: Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007)

	Ref	Expression
Assertion	cnvresid	⊢ ◡ ( I ↾ 𝐴 ) = ( I ↾ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		cnvi	⊢ ◡ I = I
2	1	eqcomi	⊢ I = ◡ I
3		funi	⊢ Fun I
4		funeq	⊢ ( I = ◡ I → ( Fun I ↔ Fun ◡ I ) )
5	3 4	mpbii	⊢ ( I = ◡ I → Fun ◡ I )
6		funcnvres	⊢ ( Fun ◡ I → ◡ ( I ↾ 𝐴 ) = ( ◡ I ↾ ( I “ 𝐴 ) ) )
7		imai	⊢ ( I “ 𝐴 ) = 𝐴
8	1 7	reseq12i	⊢ ( ◡ I ↾ ( I “ 𝐴 ) ) = ( I ↾ 𝐴 )
9	6 8	eqtrdi	⊢ ( Fun ◡ I → ◡ ( I ↾ 𝐴 ) = ( I ↾ 𝐴 ) )
10	2 5 9	mp2b	⊢ ◡ ( I ↾ 𝐴 ) = ( I ↾ 𝐴 )