Description: An alternate function value belongs to the codomain of the function, analogous to ffvelrn . (Contributed by AV, 2-Sep-2022)
Ref | Expression | ||
---|---|---|---|
Assertion | fafv2elrn | |- ( ( F : A --> B /\ C e. A ) -> ( F '''' C ) e. B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn | |- ( F : A --> B -> F Fn A ) |
|
2 | fnafv2elrn | |- ( ( F Fn A /\ C e. A ) -> ( F '''' C ) e. ran F ) |
|
3 | 1 2 | sylan | |- ( ( F : A --> B /\ C e. A ) -> ( F '''' C ) e. ran F ) |
4 | frn | |- ( F : A --> B -> ran F C_ B ) |
|
5 | 4 | sseld | |- ( F : A --> B -> ( ( F '''' C ) e. ran F -> ( F '''' C ) e. B ) ) |
6 | 5 | adantr | |- ( ( F : A --> B /\ C e. A ) -> ( ( F '''' C ) e. ran F -> ( F '''' C ) e. B ) ) |
7 | 3 6 | mpd | |- ( ( F : A --> B /\ C e. A ) -> ( F '''' C ) e. B ) |