Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrrn.t |
|- T = ( pmTrsp ` D ) |
2 |
|
pmtrrn.r |
|- R = ran T |
3 |
|
eqid |
|- dom ( F \ _I ) = dom ( F \ _I ) |
4 |
1 2 3
|
pmtrfrn |
|- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) ) |
5 |
4
|
simpld |
|- ( F e. R -> ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) ) |
6 |
5
|
simp3d |
|- ( F e. R -> dom ( F \ _I ) ~~ 2o ) |
7 |
|
en2 |
|- ( dom ( F \ _I ) ~~ 2o -> E. x E. y dom ( F \ _I ) = { x , y } ) |
8 |
6 7
|
syl |
|- ( F e. R -> E. x E. y dom ( F \ _I ) = { x , y } ) |
9 |
5
|
simp2d |
|- ( F e. R -> dom ( F \ _I ) C_ D ) |
10 |
4
|
simprd |
|- ( F e. R -> F = ( T ` dom ( F \ _I ) ) ) |
11 |
9 6 10
|
jca32 |
|- ( F e. R -> ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) ) |
12 |
|
sseq1 |
|- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) C_ D <-> { x , y } C_ D ) ) |
13 |
|
breq1 |
|- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) ~~ 2o <-> { x , y } ~~ 2o ) ) |
14 |
|
fveq2 |
|- ( dom ( F \ _I ) = { x , y } -> ( T ` dom ( F \ _I ) ) = ( T ` { x , y } ) ) |
15 |
14
|
eqeq2d |
|- ( dom ( F \ _I ) = { x , y } -> ( F = ( T ` dom ( F \ _I ) ) <-> F = ( T ` { x , y } ) ) ) |
16 |
13 15
|
anbi12d |
|- ( dom ( F \ _I ) = { x , y } -> ( ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) <-> ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) |
17 |
12 16
|
anbi12d |
|- ( dom ( F \ _I ) = { x , y } -> ( ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) <-> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) ) |
18 |
11 17
|
syl5ibcom |
|- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) ) |
19 |
|
vex |
|- x e. _V |
20 |
|
vex |
|- y e. _V |
21 |
19 20
|
prss |
|- ( ( x e. D /\ y e. D ) <-> { x , y } C_ D ) |
22 |
21
|
bicomi |
|- ( { x , y } C_ D <-> ( x e. D /\ y e. D ) ) |
23 |
|
pr2ne |
|- ( ( x e. _V /\ y e. _V ) -> ( { x , y } ~~ 2o <-> x =/= y ) ) |
24 |
23
|
el2v |
|- ( { x , y } ~~ 2o <-> x =/= y ) |
25 |
24
|
anbi1i |
|- ( ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) <-> ( x =/= y /\ F = ( T ` { x , y } ) ) ) |
26 |
22 25
|
anbi12i |
|- ( ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) <-> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) |
27 |
18 26
|
syl6ib |
|- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) ) |
28 |
27
|
2eximdv |
|- ( F e. R -> ( E. x E. y dom ( F \ _I ) = { x , y } -> E. x E. y ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) ) |
29 |
8 28
|
mpd |
|- ( F e. R -> E. x E. y ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) |
30 |
|
r2ex |
|- ( E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) <-> E. x E. y ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) |
31 |
29 30
|
sylibr |
|- ( F e. R -> E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) ) |