Metamath Proof Explorer


Theorem assintopval

Description: The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020)

Ref Expression
Assertion assintopval M V assIntOp M = o clIntOp M | o assLaw M

Proof

Step Hyp Ref Expression
1 df-assintop assIntOp = m V o clIntOp m | o assLaw m
2 fveq2 m = M clIntOp m = clIntOp M
3 breq2 m = M o assLaw m o assLaw M
4 2 3 rabeqbidv m = M o clIntOp m | o assLaw m = o clIntOp M | o assLaw M
5 elex M V M V
6 fvex clIntOp M V
7 6 rabex o clIntOp M | o assLaw M V
8 7 a1i M V o clIntOp M | o assLaw M V
9 1 4 5 8 fvmptd3 M V assIntOp M = o clIntOp M | o assLaw M