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	$$⊢F\left[\left\{A\right\}\right]=G\left[\left\{A\right\}\right]\to F\left(A\right)=G\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq1	$$⊢F\left[\left\{A\right\}\right]=G\left[\left\{A\right\}\right]\to \left(F\left[\left\{A\right\}\right]=\left\{x\right\}\leftrightarrow G\left[\left\{A\right\}\right]=\left\{x\right\}\right)$$
2	1	abbidv	$$⊢F\left[\left\{A\right\}\right]=G\left[\left\{A\right\}\right]\to \left\{x|F\left[\left\{A\right\}\right]=\left\{x\right\}\right\}=\left\{x|G\left[\left\{A\right\}\right]=\left\{x\right\}\right\}$$
3	2	unieqd	$$⊢F\left[\left\{A\right\}\right]=G\left[\left\{A\right\}\right]\to \bigcup \left\{x|F\left[\left\{A\right\}\right]=\left\{x\right\}\right\}=\bigcup \left\{x|G\left[\left\{A\right\}\right]=\left\{x\right\}\right\}$$
4		dffv4	$$⊢F\left(A\right)=\bigcup \left\{x|F\left[\left\{A\right\}\right]=\left\{x\right\}\right\}$$
5		dffv4	$$⊢G\left(A\right)=\bigcup \left\{x|G\left[\left\{A\right\}\right]=\left\{x\right\}\right\}$$
6	3 4 5	3eqtr4g	$$⊢F\left[\left\{A\right\}\right]=G\left[\left\{A\right\}\right]\to F\left(A\right)=G\left(A\right)$$