Description: The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017) (Revised by AV, 15-Jan-2021) Add disjoint variable condition on D , f , h to remove hypotheses. (Revised by SN, 13-Dec-2024)
Ref | Expression | ||
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Hypotheses | mptmpoopabbrd.g | |- ( ph -> G e. W ) |
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mptmpoopabbrd.x | |- ( ph -> X e. ( A ` G ) ) |
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mptmpoopabbrd.y | |- ( ph -> Y e. ( B ` G ) ) |
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mptmpoopabovd.m | |- M = ( g e. _V |-> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( f ( a ( C ` g ) b ) h /\ f ( D ` g ) h ) } ) ) |
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Assertion | mptmpoopabovd | |- ( ph -> ( X ( M ` G ) Y ) = { <. f , h >. | ( f ( X ( C ` G ) Y ) h /\ f ( D ` G ) h ) } ) |
Step | Hyp | Ref | Expression |
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1 | mptmpoopabbrd.g | |- ( ph -> G e. W ) |
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2 | mptmpoopabbrd.x | |- ( ph -> X e. ( A ` G ) ) |
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3 | mptmpoopabbrd.y | |- ( ph -> Y e. ( B ` G ) ) |
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4 | mptmpoopabovd.m | |- M = ( g e. _V |-> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( f ( a ( C ` g ) b ) h /\ f ( D ` g ) h ) } ) ) |
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5 | oveq12 | |- ( ( a = X /\ b = Y ) -> ( a ( C ` G ) b ) = ( X ( C ` G ) Y ) ) |
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6 | 5 | breqd | |- ( ( a = X /\ b = Y ) -> ( f ( a ( C ` G ) b ) h <-> f ( X ( C ` G ) Y ) h ) ) |
7 | fveq2 | |- ( g = G -> ( C ` g ) = ( C ` G ) ) |
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8 | 7 | oveqd | |- ( g = G -> ( a ( C ` g ) b ) = ( a ( C ` G ) b ) ) |
9 | 8 | breqd | |- ( g = G -> ( f ( a ( C ` g ) b ) h <-> f ( a ( C ` G ) b ) h ) ) |
10 | 1 2 3 6 9 4 | mptmpoopabbrd | |- ( ph -> ( X ( M ` G ) Y ) = { <. f , h >. | ( f ( X ( C ` G ) Y ) h /\ f ( D ` G ) h ) } ) |