Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004)
Ref | Expression | ||
---|---|---|---|
Assertion | f1ocnvfv | ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐶 ) = 𝐷 → ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 | ⊢ ( 𝐷 = ( 𝐹 ‘ 𝐶 ) → ( ◡ 𝐹 ‘ 𝐷 ) = ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) ) | |
2 | 1 | eqcoms | ⊢ ( ( 𝐹 ‘ 𝐶 ) = 𝐷 → ( ◡ 𝐹 ‘ 𝐷 ) = ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) ) |
3 | f1ocnvfv1 | ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) = 𝐶 ) | |
4 | 3 | eqeq2d | ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( ◡ 𝐹 ‘ 𝐷 ) = ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) ↔ ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) ) |
5 | 2 4 | syl5ib | ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐶 ) = 𝐷 → ( ◡ 𝐹 ‘ 𝐷 ) = 𝐶 ) ) |