Step |
Hyp |
Ref |
Expression |
1 |
|
fnwe2.su |
|- ( z = ( F ` x ) -> S = U ) |
2 |
|
fnwe2.t |
|- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) } |
3 |
|
fnwe2.s |
|- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } ) |
4 |
|
fnwe2.f |
|- ( ph -> ( F |` A ) : A --> B ) |
5 |
|
fnwe2.r |
|- ( ph -> R We B ) |
6 |
|
fnwe2lem3.a |
|- ( ph -> a e. A ) |
7 |
|
fnwe2lem3.b |
|- ( ph -> b e. A ) |
8 |
|
animorrl |
|- ( ( ph /\ ( F ` a ) R ( F ` b ) ) -> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) |
9 |
1 2
|
fnwe2val |
|- ( a T b <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) |
10 |
8 9
|
sylibr |
|- ( ( ph /\ ( F ` a ) R ( F ` b ) ) -> a T b ) |
11 |
10
|
3mix1d |
|- ( ( ph /\ ( F ` a ) R ( F ` b ) ) -> ( a T b \/ a = b \/ b T a ) ) |
12 |
|
simplr |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( F ` a ) = ( F ` b ) ) |
13 |
|
simpr |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> a [_ ( F ` a ) / z ]_ S b ) |
14 |
12 13
|
jca |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) |
15 |
14
|
olcd |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) |
16 |
15 9
|
sylibr |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> a T b ) |
17 |
16
|
3mix1d |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( a T b \/ a = b \/ b T a ) ) |
18 |
|
3mix2 |
|- ( a = b -> ( a T b \/ a = b \/ b T a ) ) |
19 |
18
|
adantl |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a = b ) -> ( a T b \/ a = b \/ b T a ) ) |
20 |
|
simplr |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( F ` a ) = ( F ` b ) ) |
21 |
20
|
eqcomd |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( F ` b ) = ( F ` a ) ) |
22 |
|
csbeq1 |
|- ( ( F ` a ) = ( F ` b ) -> [_ ( F ` a ) / z ]_ S = [_ ( F ` b ) / z ]_ S ) |
23 |
22
|
adantl |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> [_ ( F ` a ) / z ]_ S = [_ ( F ` b ) / z ]_ S ) |
24 |
23
|
breqd |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( b [_ ( F ` a ) / z ]_ S a <-> b [_ ( F ` b ) / z ]_ S a ) ) |
25 |
24
|
biimpa |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> b [_ ( F ` b ) / z ]_ S a ) |
26 |
21 25
|
jca |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) |
27 |
26
|
olcd |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) ) |
28 |
1 2
|
fnwe2val |
|- ( b T a <-> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) ) |
29 |
27 28
|
sylibr |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> b T a ) |
30 |
29
|
3mix3d |
|- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( a T b \/ a = b \/ b T a ) ) |
31 |
1 2 3
|
fnwe2lem1 |
|- ( ( ph /\ a e. A ) -> [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } ) |
32 |
6 31
|
mpdan |
|- ( ph -> [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } ) |
33 |
|
weso |
|- ( [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } -> [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } ) |
34 |
32 33
|
syl |
|- ( ph -> [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } ) |
35 |
34
|
adantr |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } ) |
36 |
|
fveqeq2 |
|- ( y = a -> ( ( F ` y ) = ( F ` a ) <-> ( F ` a ) = ( F ` a ) ) ) |
37 |
6
|
adantr |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> a e. A ) |
38 |
|
eqidd |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( F ` a ) = ( F ` a ) ) |
39 |
36 37 38
|
elrabd |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> a e. { y e. A | ( F ` y ) = ( F ` a ) } ) |
40 |
|
fveqeq2 |
|- ( y = b -> ( ( F ` y ) = ( F ` a ) <-> ( F ` b ) = ( F ` a ) ) ) |
41 |
7
|
adantr |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> b e. A ) |
42 |
|
simpr |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( F ` a ) = ( F ` b ) ) |
43 |
42
|
eqcomd |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( F ` b ) = ( F ` a ) ) |
44 |
40 41 43
|
elrabd |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> b e. { y e. A | ( F ` y ) = ( F ` a ) } ) |
45 |
|
solin |
|- ( ( [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } /\ ( a e. { y e. A | ( F ` y ) = ( F ` a ) } /\ b e. { y e. A | ( F ` y ) = ( F ` a ) } ) ) -> ( a [_ ( F ` a ) / z ]_ S b \/ a = b \/ b [_ ( F ` a ) / z ]_ S a ) ) |
46 |
35 39 44 45
|
syl12anc |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( a [_ ( F ` a ) / z ]_ S b \/ a = b \/ b [_ ( F ` a ) / z ]_ S a ) ) |
47 |
17 19 30 46
|
mpjao3dan |
|- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( a T b \/ a = b \/ b T a ) ) |
48 |
|
animorrl |
|- ( ( ph /\ ( F ` b ) R ( F ` a ) ) -> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) ) |
49 |
48 28
|
sylibr |
|- ( ( ph /\ ( F ` b ) R ( F ` a ) ) -> b T a ) |
50 |
49
|
3mix3d |
|- ( ( ph /\ ( F ` b ) R ( F ` a ) ) -> ( a T b \/ a = b \/ b T a ) ) |
51 |
|
weso |
|- ( R We B -> R Or B ) |
52 |
5 51
|
syl |
|- ( ph -> R Or B ) |
53 |
6
|
fvresd |
|- ( ph -> ( ( F |` A ) ` a ) = ( F ` a ) ) |
54 |
4 6
|
ffvelrnd |
|- ( ph -> ( ( F |` A ) ` a ) e. B ) |
55 |
53 54
|
eqeltrrd |
|- ( ph -> ( F ` a ) e. B ) |
56 |
7
|
fvresd |
|- ( ph -> ( ( F |` A ) ` b ) = ( F ` b ) ) |
57 |
4 7
|
ffvelrnd |
|- ( ph -> ( ( F |` A ) ` b ) e. B ) |
58 |
56 57
|
eqeltrrd |
|- ( ph -> ( F ` b ) e. B ) |
59 |
|
solin |
|- ( ( R Or B /\ ( ( F ` a ) e. B /\ ( F ` b ) e. B ) ) -> ( ( F ` a ) R ( F ` b ) \/ ( F ` a ) = ( F ` b ) \/ ( F ` b ) R ( F ` a ) ) ) |
60 |
52 55 58 59
|
syl12anc |
|- ( ph -> ( ( F ` a ) R ( F ` b ) \/ ( F ` a ) = ( F ` b ) \/ ( F ` b ) R ( F ` a ) ) ) |
61 |
11 47 50 60
|
mpjao3dan |
|- ( ph -> ( a T b \/ a = b \/ b T a ) ) |