Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrrn.t |
|- T = ( pmTrsp ` D ) |
2 |
|
pmtrrn.r |
|- R = ran T |
3 |
|
2on0 |
|- 2o =/= (/) |
4 |
|
eqid |
|- dom ( F \ _I ) = dom ( F \ _I ) |
5 |
1 2 4
|
pmtrfrn |
|- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) ) |
6 |
5
|
simpld |
|- ( F e. R -> ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) ) |
7 |
6
|
simp3d |
|- ( F e. R -> dom ( F \ _I ) ~~ 2o ) |
8 |
|
enen1 |
|- ( dom ( F \ _I ) ~~ 2o -> ( dom ( F \ _I ) ~~ (/) <-> 2o ~~ (/) ) ) |
9 |
7 8
|
syl |
|- ( F e. R -> ( dom ( F \ _I ) ~~ (/) <-> 2o ~~ (/) ) ) |
10 |
|
en0 |
|- ( dom ( F \ _I ) ~~ (/) <-> dom ( F \ _I ) = (/) ) |
11 |
|
en0 |
|- ( 2o ~~ (/) <-> 2o = (/) ) |
12 |
9 10 11
|
3bitr3g |
|- ( F e. R -> ( dom ( F \ _I ) = (/) <-> 2o = (/) ) ) |
13 |
12
|
necon3bid |
|- ( F e. R -> ( dom ( F \ _I ) =/= (/) <-> 2o =/= (/) ) ) |
14 |
3 13
|
mpbiri |
|- ( F e. R -> dom ( F \ _I ) =/= (/) ) |