Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrdifel.t |
|- T = ran ( pmTrsp ` ( N \ { K } ) ) |
2 |
|
pmtrdifel.r |
|- R = ran ( pmTrsp ` N ) |
3 |
|
pmtrdifwrdel.0 |
|- U = ( x e. ( 0 ..^ ( # ` W ) ) |-> ( ( pmTrsp ` N ) ` dom ( ( W ` x ) \ _I ) ) ) |
4 |
|
simpr |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> i e. ( 0 ..^ ( # ` W ) ) ) |
5 |
|
fvex |
|- ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) e. _V |
6 |
|
fveq2 |
|- ( x = i -> ( W ` x ) = ( W ` i ) ) |
7 |
6
|
difeq1d |
|- ( x = i -> ( ( W ` x ) \ _I ) = ( ( W ` i ) \ _I ) ) |
8 |
7
|
dmeqd |
|- ( x = i -> dom ( ( W ` x ) \ _I ) = dom ( ( W ` i ) \ _I ) ) |
9 |
8
|
fveq2d |
|- ( x = i -> ( ( pmTrsp ` N ) ` dom ( ( W ` x ) \ _I ) ) = ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) ) |
10 |
9 3
|
fvmptg |
|- ( ( i e. ( 0 ..^ ( # ` W ) ) /\ ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) e. _V ) -> ( U ` i ) = ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) ) |
11 |
4 5 10
|
sylancl |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( U ` i ) = ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) ) |
12 |
11
|
fveq1d |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( ( U ` i ) ` K ) = ( ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) ` K ) ) |
13 |
|
wrdsymbcl |
|- ( ( W e. Word T /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` i ) e. T ) |
14 |
13
|
adantlr |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` i ) e. T ) |
15 |
|
simplr |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> K e. N ) |
16 |
|
eqid |
|- ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) = ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) |
17 |
1 2 16
|
pmtrdifellem4 |
|- ( ( ( W ` i ) e. T /\ K e. N ) -> ( ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) ` K ) = K ) |
18 |
14 15 17
|
syl2anc |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( pmTrsp ` N ) ` dom ( ( W ` i ) \ _I ) ) ` K ) = K ) |
19 |
12 18
|
eqtrd |
|- ( ( ( W e. Word T /\ K e. N ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( ( U ` i ) ` K ) = K ) |
20 |
19
|
ralrimiva |
|- ( ( W e. Word T /\ K e. N ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( ( U ` i ) ` K ) = K ) |