Theorem f1omvdcnv

Description: A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015)

	Ref	Expression
Assertion	f1omvdcnv	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → dom ( ◡ 𝐹 ∖ I ) = dom ( 𝐹 ∖ I ) )

Proof

Step	Hyp	Ref	Expression
1		f1ocnvfvb	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝑥 ↔ ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ) )
2	1	3anidm23	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝑥 ↔ ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ) )
3	2	bicomd	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ↔ ( 𝐹 ‘ 𝑥 ) = 𝑥 ) )
4	3	necon3bid	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( ◡ 𝐹 ‘ 𝑥 ) ≠ 𝑥 ↔ ( 𝐹 ‘ 𝑥 ) ≠ 𝑥 ) )
5	4	rabbidva	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → { 𝑥 ∈ 𝐴 ∣ ( ◡ 𝐹 ‘ 𝑥 ) ≠ 𝑥 } = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑥 } )
6		f1ocnv	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → ◡ 𝐹 : 𝐴 –1-1-onto→ 𝐴 )
7		f1ofn	⊢ ( ◡ 𝐹 : 𝐴 –1-1-onto→ 𝐴 → ◡ 𝐹 Fn 𝐴 )
8		fndifnfp	⊢ ( ◡ 𝐹 Fn 𝐴 → dom ( ◡ 𝐹 ∖ I ) = { 𝑥 ∈ 𝐴 ∣ ( ◡ 𝐹 ‘ 𝑥 ) ≠ 𝑥 } )
9	6 7 8	3syl	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → dom ( ◡ 𝐹 ∖ I ) = { 𝑥 ∈ 𝐴 ∣ ( ◡ 𝐹 ‘ 𝑥 ) ≠ 𝑥 } )
10		f1ofn	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → 𝐹 Fn 𝐴 )
11		fndifnfp	⊢ ( 𝐹 Fn 𝐴 → dom ( 𝐹 ∖ I ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑥 } )
12	10 11	syl	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → dom ( 𝐹 ∖ I ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑥 } )
13	5 9 12	3eqtr4d	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → dom ( ◡ 𝐹 ∖ I ) = dom ( 𝐹 ∖ I ) )