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Metamath Proof Explorer

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 ) )