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Metamath Proof Explorer

Theorem f1ocnvfvb

Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004)

Ref Expression
Assertion f1ocnvfvb ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝐶 ) = 𝐷 ↔ ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 f1ocnvfv ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐶 ) = 𝐷 → ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) )
2 1 3adant3 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝐶 ) = 𝐷 → ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) )
3 fveq2 ⊢ ( 𝐶 = ( ◡ 𝐹 ‘ 𝐷 ) → ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) )
4 3 eqcoms ⊢ ( ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 → ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) )
5 f1ocnvfv2 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) = 𝐷 )
6 5 eqeq2d ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) ↔ ( 𝐹 ‘ 𝐶 ) = 𝐷 ) )
7 4 6 imbitrid ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 → ( 𝐹 ‘ 𝐶 ) = 𝐷 ) )
8 7 3adant2 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 → ( 𝐹 ‘ 𝐶 ) = 𝐷 ) )
9 2 8 impbid ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝐶 ) = 𝐷 ↔ ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) )