Theorem bj-imafv

Description: If the direct image of a singleton under any of two functions is the same, then the values of these functions at the corresponding point agree. (Contributed by BJ, 18-Mar-2023)

	Ref	Expression
Assertion	bj-imafv	⊢ ( ( 𝐹 “ { 𝐴 } ) = ( 𝐺 “ { 𝐴 } ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	⊢ ( ( 𝐹 “ { 𝐴 } ) = ( 𝐺 “ { 𝐴 } ) → ( ( 𝐹 “ { 𝐴 } ) = { 𝑥 } ↔ ( 𝐺 “ { 𝐴 } ) = { 𝑥 } ) )
2	1	abbidv	⊢ ( ( 𝐹 “ { 𝐴 } ) = ( 𝐺 “ { 𝐴 } ) → { 𝑥 ∣ ( 𝐹 “ { 𝐴 } ) = { 𝑥 } } = { 𝑥 ∣ ( 𝐺 “ { 𝐴 } ) = { 𝑥 } } )
3	2	unieqd	⊢ ( ( 𝐹 “ { 𝐴 } ) = ( 𝐺 “ { 𝐴 } ) → ∪ { 𝑥 ∣ ( 𝐹 “ { 𝐴 } ) = { 𝑥 } } = ∪ { 𝑥 ∣ ( 𝐺 “ { 𝐴 } ) = { 𝑥 } } )
4		dffv4	⊢ ( 𝐹 ‘ 𝐴 ) = ∪ { 𝑥 ∣ ( 𝐹 “ { 𝐴 } ) = { 𝑥 } }
5		dffv4	⊢ ( 𝐺 ‘ 𝐴 ) = ∪ { 𝑥 ∣ ( 𝐺 “ { 𝐴 } ) = { 𝑥 } }
6	3 4 5	3eqtr4g	⊢ ( ( 𝐹 “ { 𝐴 } ) = ( 𝐺 “ { 𝐴 } ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )