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 ) )