Step |
Hyp |
Ref |
Expression |
1 |
|
fsuppcurry1.g |
|- G = ( x e. B |-> ( C F x ) ) |
2 |
|
fsuppcurry1.z |
|- ( ph -> Z e. U ) |
3 |
|
fsuppcurry1.a |
|- ( ph -> A e. V ) |
4 |
|
fsuppcurry1.b |
|- ( ph -> B e. W ) |
5 |
|
fsuppcurry1.f |
|- ( ph -> F Fn ( A X. B ) ) |
6 |
|
fsuppcurry1.c |
|- ( ph -> C e. A ) |
7 |
|
fsuppcurry1.1 |
|- ( ph -> F finSupp Z ) |
8 |
|
oveq2 |
|- ( x = y -> ( C F x ) = ( C F y ) ) |
9 |
8
|
cbvmptv |
|- ( x e. B |-> ( C F x ) ) = ( y e. B |-> ( C F y ) ) |
10 |
1 9
|
eqtri |
|- G = ( y e. B |-> ( C F y ) ) |
11 |
4
|
mptexd |
|- ( ph -> ( y e. B |-> ( C F y ) ) e. _V ) |
12 |
10 11
|
eqeltrid |
|- ( ph -> G e. _V ) |
13 |
1
|
funmpt2 |
|- Fun G |
14 |
13
|
a1i |
|- ( ph -> Fun G ) |
15 |
|
fo2nd |
|- 2nd : _V -onto-> _V |
16 |
|
fofun |
|- ( 2nd : _V -onto-> _V -> Fun 2nd ) |
17 |
15 16
|
ax-mp |
|- Fun 2nd |
18 |
|
funres |
|- ( Fun 2nd -> Fun ( 2nd |` ( _V X. _V ) ) ) |
19 |
17 18
|
mp1i |
|- ( ph -> Fun ( 2nd |` ( _V X. _V ) ) ) |
20 |
7
|
fsuppimpd |
|- ( ph -> ( F supp Z ) e. Fin ) |
21 |
|
imafi |
|- ( ( Fun ( 2nd |` ( _V X. _V ) ) /\ ( F supp Z ) e. Fin ) -> ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) e. Fin ) |
22 |
19 20 21
|
syl2anc |
|- ( ph -> ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) e. Fin ) |
23 |
|
ovexd |
|- ( ( ph /\ y e. B ) -> ( C F y ) e. _V ) |
24 |
23 10
|
fmptd |
|- ( ph -> G : B --> _V ) |
25 |
|
eldif |
|- ( y e. ( B \ ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) <-> ( y e. B /\ -. y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) ) |
26 |
6
|
ad2antrr |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> C e. A ) |
27 |
|
simplr |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> y e. B ) |
28 |
26 27
|
opelxpd |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> <. C , y >. e. ( A X. B ) ) |
29 |
|
df-ov |
|- ( C F y ) = ( F ` <. C , y >. ) |
30 |
|
ovexd |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( C F y ) e. _V ) |
31 |
1 8 27 30
|
fvmptd3 |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( G ` y ) = ( C F y ) ) |
32 |
|
simpr |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> -. ( G ` y ) = Z ) |
33 |
32
|
neqned |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( G ` y ) =/= Z ) |
34 |
31 33
|
eqnetrrd |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( C F y ) =/= Z ) |
35 |
29 34
|
eqnetrrid |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( F ` <. C , y >. ) =/= Z ) |
36 |
3 4
|
xpexd |
|- ( ph -> ( A X. B ) e. _V ) |
37 |
|
elsuppfn |
|- ( ( F Fn ( A X. B ) /\ ( A X. B ) e. _V /\ Z e. U ) -> ( <. C , y >. e. ( F supp Z ) <-> ( <. C , y >. e. ( A X. B ) /\ ( F ` <. C , y >. ) =/= Z ) ) ) |
38 |
5 36 2 37
|
syl3anc |
|- ( ph -> ( <. C , y >. e. ( F supp Z ) <-> ( <. C , y >. e. ( A X. B ) /\ ( F ` <. C , y >. ) =/= Z ) ) ) |
39 |
38
|
ad2antrr |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( <. C , y >. e. ( F supp Z ) <-> ( <. C , y >. e. ( A X. B ) /\ ( F ` <. C , y >. ) =/= Z ) ) ) |
40 |
28 35 39
|
mpbir2and |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> <. C , y >. e. ( F supp Z ) ) |
41 |
|
simpr |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> z = <. C , y >. ) |
42 |
41
|
fveq2d |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> ( ( 2nd |` ( _V X. _V ) ) ` z ) = ( ( 2nd |` ( _V X. _V ) ) ` <. C , y >. ) ) |
43 |
|
xpss |
|- ( A X. B ) C_ ( _V X. _V ) |
44 |
28
|
adantr |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> <. C , y >. e. ( A X. B ) ) |
45 |
43 44
|
sselid |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> <. C , y >. e. ( _V X. _V ) ) |
46 |
45
|
fvresd |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> ( ( 2nd |` ( _V X. _V ) ) ` <. C , y >. ) = ( 2nd ` <. C , y >. ) ) |
47 |
26
|
adantr |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> C e. A ) |
48 |
27
|
adantr |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> y e. B ) |
49 |
|
op2ndg |
|- ( ( C e. A /\ y e. B ) -> ( 2nd ` <. C , y >. ) = y ) |
50 |
47 48 49
|
syl2anc |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> ( 2nd ` <. C , y >. ) = y ) |
51 |
42 46 50
|
3eqtrd |
|- ( ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) /\ z = <. C , y >. ) -> ( ( 2nd |` ( _V X. _V ) ) ` z ) = y ) |
52 |
40 51
|
rspcedeq1vd |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> E. z e. ( F supp Z ) ( ( 2nd |` ( _V X. _V ) ) ` z ) = y ) |
53 |
|
fofn |
|- ( 2nd : _V -onto-> _V -> 2nd Fn _V ) |
54 |
|
fnresin |
|- ( 2nd Fn _V -> ( 2nd |` ( _V X. _V ) ) Fn ( _V i^i ( _V X. _V ) ) ) |
55 |
15 53 54
|
mp2b |
|- ( 2nd |` ( _V X. _V ) ) Fn ( _V i^i ( _V X. _V ) ) |
56 |
|
ssv |
|- ( _V X. _V ) C_ _V |
57 |
|
sseqin2 |
|- ( ( _V X. _V ) C_ _V <-> ( _V i^i ( _V X. _V ) ) = ( _V X. _V ) ) |
58 |
56 57
|
mpbi |
|- ( _V i^i ( _V X. _V ) ) = ( _V X. _V ) |
59 |
58
|
fneq2i |
|- ( ( 2nd |` ( _V X. _V ) ) Fn ( _V i^i ( _V X. _V ) ) <-> ( 2nd |` ( _V X. _V ) ) Fn ( _V X. _V ) ) |
60 |
55 59
|
mpbi |
|- ( 2nd |` ( _V X. _V ) ) Fn ( _V X. _V ) |
61 |
60
|
a1i |
|- ( ph -> ( 2nd |` ( _V X. _V ) ) Fn ( _V X. _V ) ) |
62 |
|
suppssdm |
|- ( F supp Z ) C_ dom F |
63 |
5
|
fndmd |
|- ( ph -> dom F = ( A X. B ) ) |
64 |
62 63
|
sseqtrid |
|- ( ph -> ( F supp Z ) C_ ( A X. B ) ) |
65 |
64 43
|
sstrdi |
|- ( ph -> ( F supp Z ) C_ ( _V X. _V ) ) |
66 |
61 65
|
fvelimabd |
|- ( ph -> ( y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) <-> E. z e. ( F supp Z ) ( ( 2nd |` ( _V X. _V ) ) ` z ) = y ) ) |
67 |
66
|
ad2antrr |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> ( y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) <-> E. z e. ( F supp Z ) ( ( 2nd |` ( _V X. _V ) ) ` z ) = y ) ) |
68 |
52 67
|
mpbird |
|- ( ( ( ph /\ y e. B ) /\ -. ( G ` y ) = Z ) -> y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) |
69 |
68
|
ex |
|- ( ( ph /\ y e. B ) -> ( -. ( G ` y ) = Z -> y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) ) |
70 |
69
|
con1d |
|- ( ( ph /\ y e. B ) -> ( -. y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) -> ( G ` y ) = Z ) ) |
71 |
70
|
impr |
|- ( ( ph /\ ( y e. B /\ -. y e. ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) ) -> ( G ` y ) = Z ) |
72 |
25 71
|
sylan2b |
|- ( ( ph /\ y e. ( B \ ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) ) -> ( G ` y ) = Z ) |
73 |
24 72
|
suppss |
|- ( ph -> ( G supp Z ) C_ ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) |
74 |
|
suppssfifsupp |
|- ( ( ( G e. _V /\ Fun G /\ Z e. U ) /\ ( ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) e. Fin /\ ( G supp Z ) C_ ( ( 2nd |` ( _V X. _V ) ) " ( F supp Z ) ) ) ) -> G finSupp Z ) |
75 |
12 14 2 22 73 74
|
syl32anc |
|- ( ph -> G finSupp Z ) |