Step |
Hyp |
Ref |
Expression |
1 |
|
seqom.a |
|- G = seqom ( F , I ) |
2 |
|
seqomlem0 |
|- rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) = rec ( ( c e. _om , d e. _V |-> <. suc c , ( c F d ) >. ) , <. (/) , ( _I ` I ) >. ) |
3 |
2
|
seqomlem4 |
|- ( A e. _om -> ( ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) ` suc A ) = ( A F ( ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) ` A ) ) ) |
4 |
|
df-seqom |
|- seqom ( F , I ) = ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) |
5 |
1 4
|
eqtri |
|- G = ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) |
6 |
5
|
fveq1i |
|- ( G ` suc A ) = ( ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) ` suc A ) |
7 |
5
|
fveq1i |
|- ( G ` A ) = ( ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) ` A ) |
8 |
7
|
oveq2i |
|- ( A F ( G ` A ) ) = ( A F ( ( rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) ` A ) ) |
9 |
3 6 8
|
3eqtr4g |
|- ( A e. _om -> ( G ` suc A ) = ( A F ( G ` A ) ) ) |