Step |
Hyp |
Ref |
Expression |
1 |
|
qsalrel.1 |
|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> x .~ y ) |
2 |
|
qsalrel.2 |
|- ( ph -> .~ Er A ) |
3 |
|
qsalrel.3 |
|- ( ph -> N e. A ) |
4 |
|
dfqs2 |
|- ( A /. .~ ) = ran ( a e. A |-> [ a ] .~ ) |
5 |
2
|
adantr |
|- ( ( ph /\ a e. A ) -> .~ Er A ) |
6 |
1
|
ralrimivva |
|- ( ph -> A. x e. A A. y e. A x .~ y ) |
7 |
6
|
adantr |
|- ( ( ph /\ a e. A ) -> A. x e. A A. y e. A x .~ y ) |
8 |
|
simpr |
|- ( ( ph /\ a e. A ) -> a e. A ) |
9 |
|
breq1 |
|- ( x = a -> ( x .~ y <-> a .~ y ) ) |
10 |
9
|
ralbidv |
|- ( x = a -> ( A. y e. A x .~ y <-> A. y e. A a .~ y ) ) |
11 |
10
|
adantl |
|- ( ( ( ph /\ a e. A ) /\ x = a ) -> ( A. y e. A x .~ y <-> A. y e. A a .~ y ) ) |
12 |
8 11
|
rspcdv |
|- ( ( ph /\ a e. A ) -> ( A. x e. A A. y e. A x .~ y -> A. y e. A a .~ y ) ) |
13 |
|
breq2 |
|- ( y = N -> ( a .~ y <-> a .~ N ) ) |
14 |
13
|
adantl |
|- ( ( ph /\ y = N ) -> ( a .~ y <-> a .~ N ) ) |
15 |
3 14
|
rspcdv |
|- ( ph -> ( A. y e. A a .~ y -> a .~ N ) ) |
16 |
15
|
adantr |
|- ( ( ph /\ a e. A ) -> ( A. y e. A a .~ y -> a .~ N ) ) |
17 |
12 16
|
syld |
|- ( ( ph /\ a e. A ) -> ( A. x e. A A. y e. A x .~ y -> a .~ N ) ) |
18 |
7 17
|
mpd |
|- ( ( ph /\ a e. A ) -> a .~ N ) |
19 |
5 18
|
erthi |
|- ( ( ph /\ a e. A ) -> [ a ] .~ = [ N ] .~ ) |
20 |
19
|
mpteq2dva |
|- ( ph -> ( a e. A |-> [ a ] .~ ) = ( a e. A |-> [ N ] .~ ) ) |
21 |
20
|
rneqd |
|- ( ph -> ran ( a e. A |-> [ a ] .~ ) = ran ( a e. A |-> [ N ] .~ ) ) |
22 |
|
eqid |
|- ( a e. A |-> [ N ] .~ ) = ( a e. A |-> [ N ] .~ ) |
23 |
3
|
ne0d |
|- ( ph -> A =/= (/) ) |
24 |
22 23
|
rnmptc |
|- ( ph -> ran ( a e. A |-> [ N ] .~ ) = { [ N ] .~ } ) |
25 |
2
|
ecss |
|- ( ph -> [ N ] .~ C_ A ) |
26 |
5 18
|
ersym |
|- ( ( ph /\ a e. A ) -> N .~ a ) |
27 |
3
|
adantr |
|- ( ( ph /\ a e. A ) -> N e. A ) |
28 |
|
elecg |
|- ( ( a e. A /\ N e. A ) -> ( a e. [ N ] .~ <-> N .~ a ) ) |
29 |
8 27 28
|
syl2anc |
|- ( ( ph /\ a e. A ) -> ( a e. [ N ] .~ <-> N .~ a ) ) |
30 |
26 29
|
mpbird |
|- ( ( ph /\ a e. A ) -> a e. [ N ] .~ ) |
31 |
25 30
|
eqelssd |
|- ( ph -> [ N ] .~ = A ) |
32 |
31
|
sneqd |
|- ( ph -> { [ N ] .~ } = { A } ) |
33 |
21 24 32
|
3eqtrd |
|- ( ph -> ran ( a e. A |-> [ a ] .~ ) = { A } ) |
34 |
4 33
|
syl5eq |
|- ( ph -> ( A /. .~ ) = { A } ) |