Description: The value of a function that maps from B to A . (Contributed by AV, 2-Feb-2023)
Ref | Expression | ||
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Hypotheses | mapfvd.m | |- M = ( A ^m B ) |
|
mapfvd.f | |- ( ph -> F e. M ) |
||
mapfvd.x | |- ( ph -> X e. B ) |
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Assertion | mapfvd | |- ( ph -> ( F ` X ) e. A ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfvd.m | |- M = ( A ^m B ) |
|
2 | mapfvd.f | |- ( ph -> F e. M ) |
|
3 | mapfvd.x | |- ( ph -> X e. B ) |
|
4 | elmapi | |- ( F e. ( A ^m B ) -> F : B --> A ) |
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5 | ffvelrn | |- ( ( F : B --> A /\ X e. B ) -> ( F ` X ) e. A ) |
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6 | 5 | expcom | |- ( X e. B -> ( F : B --> A -> ( F ` X ) e. A ) ) |
7 | 3 4 6 | syl2imc | |- ( F e. ( A ^m B ) -> ( ph -> ( F ` X ) e. A ) ) |
8 | 7 1 | eleq2s | |- ( F e. M -> ( ph -> ( F ` X ) e. A ) ) |
9 | 2 8 | mpcom | |- ( ph -> ( F ` X ) e. A ) |