Description: Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | fvmap.a | |- ( ph -> A e. V ) |
|
fvmap.b | |- ( ph -> B e. W ) |
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fvmap.f | |- ( ph -> F e. ( A ^m B ) ) |
||
fvmap.c | |- ( ph -> C e. B ) |
||
Assertion | fvmap | |- ( ph -> ( F ` C ) e. A ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmap.a | |- ( ph -> A e. V ) |
|
2 | fvmap.b | |- ( ph -> B e. W ) |
|
3 | fvmap.f | |- ( ph -> F e. ( A ^m B ) ) |
|
4 | fvmap.c | |- ( ph -> C e. B ) |
|
5 | id | |- ( ph -> ph ) |
|
6 | elmapg | |- ( ( A e. V /\ B e. W ) -> ( F e. ( A ^m B ) <-> F : B --> A ) ) |
|
7 | 1 2 6 | syl2anc | |- ( ph -> ( F e. ( A ^m B ) <-> F : B --> A ) ) |
8 | 3 7 | mpbid | |- ( ph -> F : B --> A ) |
9 | 8 | ffvelrnda | |- ( ( ph /\ C e. B ) -> ( F ` C ) e. A ) |
10 | 5 4 9 | syl2anc | |- ( ph -> ( F ` C ) e. A ) |