Step |
Hyp |
Ref |
Expression |
1 |
|
pmtridf1o.a |
|- ( ph -> A e. V ) |
2 |
|
pmtridf1o.x |
|- ( ph -> X e. A ) |
3 |
|
pmtridf1o.y |
|- ( ph -> Y e. A ) |
4 |
|
pmtridf1o.t |
|- T = if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) |
5 |
|
iftrue |
|- ( X = Y -> if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) = ( _I |` A ) ) |
6 |
5
|
adantl |
|- ( ( ph /\ X = Y ) -> if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) = ( _I |` A ) ) |
7 |
4 6
|
syl5eq |
|- ( ( ph /\ X = Y ) -> T = ( _I |` A ) ) |
8 |
|
f1oi |
|- ( _I |` A ) : A -1-1-onto-> A |
9 |
8
|
a1i |
|- ( ( ph /\ X = Y ) -> ( _I |` A ) : A -1-1-onto-> A ) |
10 |
|
f1oeq1 |
|- ( T = ( _I |` A ) -> ( T : A -1-1-onto-> A <-> ( _I |` A ) : A -1-1-onto-> A ) ) |
11 |
10
|
biimpar |
|- ( ( T = ( _I |` A ) /\ ( _I |` A ) : A -1-1-onto-> A ) -> T : A -1-1-onto-> A ) |
12 |
7 9 11
|
syl2anc |
|- ( ( ph /\ X = Y ) -> T : A -1-1-onto-> A ) |
13 |
|
simpr |
|- ( ( ph /\ X =/= Y ) -> X =/= Y ) |
14 |
13
|
neneqd |
|- ( ( ph /\ X =/= Y ) -> -. X = Y ) |
15 |
|
iffalse |
|- ( -. X = Y -> if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) = ( ( pmTrsp ` A ) ` { X , Y } ) ) |
16 |
14 15
|
syl |
|- ( ( ph /\ X =/= Y ) -> if ( X = Y , ( _I |` A ) , ( ( pmTrsp ` A ) ` { X , Y } ) ) = ( ( pmTrsp ` A ) ` { X , Y } ) ) |
17 |
4 16
|
syl5eq |
|- ( ( ph /\ X =/= Y ) -> T = ( ( pmTrsp ` A ) ` { X , Y } ) ) |
18 |
1
|
adantr |
|- ( ( ph /\ X =/= Y ) -> A e. V ) |
19 |
2
|
adantr |
|- ( ( ph /\ X =/= Y ) -> X e. A ) |
20 |
3
|
adantr |
|- ( ( ph /\ X =/= Y ) -> Y e. A ) |
21 |
19 20
|
prssd |
|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ A ) |
22 |
|
pr2nelem |
|- ( ( X e. A /\ Y e. A /\ X =/= Y ) -> { X , Y } ~~ 2o ) |
23 |
19 20 13 22
|
syl3anc |
|- ( ( ph /\ X =/= Y ) -> { X , Y } ~~ 2o ) |
24 |
|
eqid |
|- ( pmTrsp ` A ) = ( pmTrsp ` A ) |
25 |
|
eqid |
|- ran ( pmTrsp ` A ) = ran ( pmTrsp ` A ) |
26 |
24 25
|
pmtrrn |
|- ( ( A e. V /\ { X , Y } C_ A /\ { X , Y } ~~ 2o ) -> ( ( pmTrsp ` A ) ` { X , Y } ) e. ran ( pmTrsp ` A ) ) |
27 |
18 21 23 26
|
syl3anc |
|- ( ( ph /\ X =/= Y ) -> ( ( pmTrsp ` A ) ` { X , Y } ) e. ran ( pmTrsp ` A ) ) |
28 |
17 27
|
eqeltrd |
|- ( ( ph /\ X =/= Y ) -> T e. ran ( pmTrsp ` A ) ) |
29 |
24 25
|
pmtrff1o |
|- ( T e. ran ( pmTrsp ` A ) -> T : A -1-1-onto-> A ) |
30 |
28 29
|
syl |
|- ( ( ph /\ X =/= Y ) -> T : A -1-1-onto-> A ) |
31 |
12 30
|
pm2.61dane |
|- ( ph -> T : A -1-1-onto-> A ) |