Step |
Hyp |
Ref |
Expression |
1 |
|
isssc.1 |
|- ( ph -> H Fn ( S X. S ) ) |
2 |
|
isssc.2 |
|- ( ph -> J Fn ( T X. T ) ) |
3 |
|
isssc.3 |
|- ( ph -> T e. V ) |
4 |
|
brssc |
|- ( H C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) |
5 |
|
fndm |
|- ( J Fn ( t X. t ) -> dom J = ( t X. t ) ) |
6 |
5
|
adantl |
|- ( ( ph /\ J Fn ( t X. t ) ) -> dom J = ( t X. t ) ) |
7 |
2
|
adantr |
|- ( ( ph /\ J Fn ( t X. t ) ) -> J Fn ( T X. T ) ) |
8 |
7
|
fndmd |
|- ( ( ph /\ J Fn ( t X. t ) ) -> dom J = ( T X. T ) ) |
9 |
6 8
|
eqtr3d |
|- ( ( ph /\ J Fn ( t X. t ) ) -> ( t X. t ) = ( T X. T ) ) |
10 |
9
|
dmeqd |
|- ( ( ph /\ J Fn ( t X. t ) ) -> dom ( t X. t ) = dom ( T X. T ) ) |
11 |
|
dmxpid |
|- dom ( t X. t ) = t |
12 |
|
dmxpid |
|- dom ( T X. T ) = T |
13 |
10 11 12
|
3eqtr3g |
|- ( ( ph /\ J Fn ( t X. t ) ) -> t = T ) |
14 |
13
|
ex |
|- ( ph -> ( J Fn ( t X. t ) -> t = T ) ) |
15 |
|
id |
|- ( t = T -> t = T ) |
16 |
15
|
sqxpeqd |
|- ( t = T -> ( t X. t ) = ( T X. T ) ) |
17 |
16
|
fneq2d |
|- ( t = T -> ( J Fn ( t X. t ) <-> J Fn ( T X. T ) ) ) |
18 |
2 17
|
syl5ibrcom |
|- ( ph -> ( t = T -> J Fn ( t X. t ) ) ) |
19 |
14 18
|
impbid |
|- ( ph -> ( J Fn ( t X. t ) <-> t = T ) ) |
20 |
19
|
anbi1d |
|- ( ph -> ( ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> ( t = T /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) ) |
21 |
20
|
exbidv |
|- ( ph -> ( E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> E. t ( t = T /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) ) |
22 |
4 21
|
syl5bb |
|- ( ph -> ( H C_cat J <-> E. t ( t = T /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) ) |
23 |
|
pweq |
|- ( t = T -> ~P t = ~P T ) |
24 |
23
|
rexeqdv |
|- ( t = T -> ( E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) <-> E. s e. ~P T H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) |
25 |
24
|
ceqsexgv |
|- ( T e. V -> ( E. t ( t = T /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> E. s e. ~P T H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) |
26 |
3 25
|
syl |
|- ( ph -> ( E. t ( t = T /\ E. s e. ~P t H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> E. s e. ~P T H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) |
27 |
22 26
|
bitrd |
|- ( ph -> ( H C_cat J <-> E. s e. ~P T H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) |
28 |
|
df-rex |
|- ( E. s e. ~P T H e. X_ z e. ( s X. s ) ~P ( J ` z ) <-> E. s ( s e. ~P T /\ H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) ) |
29 |
|
3anass |
|- ( ( H e. _V /\ H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) <-> ( H e. _V /\ ( H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) |
30 |
|
elixp2 |
|- ( H e. X_ z e. ( s X. s ) ~P ( J ` z ) <-> ( H e. _V /\ H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) |
31 |
|
vex |
|- s e. _V |
32 |
31 31
|
xpex |
|- ( s X. s ) e. _V |
33 |
|
fnex |
|- ( ( H Fn ( s X. s ) /\ ( s X. s ) e. _V ) -> H e. _V ) |
34 |
32 33
|
mpan2 |
|- ( H Fn ( s X. s ) -> H e. _V ) |
35 |
34
|
adantr |
|- ( ( H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) -> H e. _V ) |
36 |
35
|
pm4.71ri |
|- ( ( H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) <-> ( H e. _V /\ ( H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) |
37 |
29 30 36
|
3bitr4i |
|- ( H e. X_ z e. ( s X. s ) ~P ( J ` z ) <-> ( H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) |
38 |
|
fndm |
|- ( H Fn ( s X. s ) -> dom H = ( s X. s ) ) |
39 |
38
|
adantl |
|- ( ( ph /\ H Fn ( s X. s ) ) -> dom H = ( s X. s ) ) |
40 |
1
|
adantr |
|- ( ( ph /\ H Fn ( s X. s ) ) -> H Fn ( S X. S ) ) |
41 |
40
|
fndmd |
|- ( ( ph /\ H Fn ( s X. s ) ) -> dom H = ( S X. S ) ) |
42 |
39 41
|
eqtr3d |
|- ( ( ph /\ H Fn ( s X. s ) ) -> ( s X. s ) = ( S X. S ) ) |
43 |
42
|
dmeqd |
|- ( ( ph /\ H Fn ( s X. s ) ) -> dom ( s X. s ) = dom ( S X. S ) ) |
44 |
|
dmxpid |
|- dom ( s X. s ) = s |
45 |
|
dmxpid |
|- dom ( S X. S ) = S |
46 |
43 44 45
|
3eqtr3g |
|- ( ( ph /\ H Fn ( s X. s ) ) -> s = S ) |
47 |
46
|
ex |
|- ( ph -> ( H Fn ( s X. s ) -> s = S ) ) |
48 |
|
id |
|- ( s = S -> s = S ) |
49 |
48
|
sqxpeqd |
|- ( s = S -> ( s X. s ) = ( S X. S ) ) |
50 |
49
|
fneq2d |
|- ( s = S -> ( H Fn ( s X. s ) <-> H Fn ( S X. S ) ) ) |
51 |
1 50
|
syl5ibrcom |
|- ( ph -> ( s = S -> H Fn ( s X. s ) ) ) |
52 |
47 51
|
impbid |
|- ( ph -> ( H Fn ( s X. s ) <-> s = S ) ) |
53 |
52
|
anbi1d |
|- ( ph -> ( ( H Fn ( s X. s ) /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) <-> ( s = S /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) |
54 |
37 53
|
syl5bb |
|- ( ph -> ( H e. X_ z e. ( s X. s ) ~P ( J ` z ) <-> ( s = S /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) |
55 |
54
|
anbi2d |
|- ( ph -> ( ( s e. ~P T /\ H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> ( s e. ~P T /\ ( s = S /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) ) |
56 |
|
an12 |
|- ( ( s e. ~P T /\ ( s = S /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) <-> ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) |
57 |
55 56
|
bitrdi |
|- ( ph -> ( ( s e. ~P T /\ H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) ) |
58 |
57
|
exbidv |
|- ( ph -> ( E. s ( s e. ~P T /\ H e. X_ z e. ( s X. s ) ~P ( J ` z ) ) <-> E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) ) |
59 |
28 58
|
syl5bb |
|- ( ph -> ( E. s e. ~P T H e. X_ z e. ( s X. s ) ~P ( J ` z ) <-> E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) ) ) |
60 |
|
exsimpl |
|- ( E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) -> E. s s = S ) |
61 |
|
isset |
|- ( S e. _V <-> E. s s = S ) |
62 |
60 61
|
sylibr |
|- ( E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) -> S e. _V ) |
63 |
62
|
a1i |
|- ( ph -> ( E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) -> S e. _V ) ) |
64 |
|
ssexg |
|- ( ( S C_ T /\ T e. V ) -> S e. _V ) |
65 |
64
|
expcom |
|- ( T e. V -> ( S C_ T -> S e. _V ) ) |
66 |
3 65
|
syl |
|- ( ph -> ( S C_ T -> S e. _V ) ) |
67 |
66
|
adantrd |
|- ( ph -> ( ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) -> S e. _V ) ) |
68 |
31
|
elpw |
|- ( s e. ~P T <-> s C_ T ) |
69 |
|
sseq1 |
|- ( s = S -> ( s C_ T <-> S C_ T ) ) |
70 |
68 69
|
syl5bb |
|- ( s = S -> ( s e. ~P T <-> S C_ T ) ) |
71 |
49
|
raleqdv |
|- ( s = S -> ( A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) <-> A. z e. ( S X. S ) ( H ` z ) e. ~P ( J ` z ) ) ) |
72 |
|
fvex |
|- ( H ` z ) e. _V |
73 |
72
|
elpw |
|- ( ( H ` z ) e. ~P ( J ` z ) <-> ( H ` z ) C_ ( J ` z ) ) |
74 |
|
fveq2 |
|- ( z = <. x , y >. -> ( H ` z ) = ( H ` <. x , y >. ) ) |
75 |
|
df-ov |
|- ( x H y ) = ( H ` <. x , y >. ) |
76 |
74 75
|
eqtr4di |
|- ( z = <. x , y >. -> ( H ` z ) = ( x H y ) ) |
77 |
|
fveq2 |
|- ( z = <. x , y >. -> ( J ` z ) = ( J ` <. x , y >. ) ) |
78 |
|
df-ov |
|- ( x J y ) = ( J ` <. x , y >. ) |
79 |
77 78
|
eqtr4di |
|- ( z = <. x , y >. -> ( J ` z ) = ( x J y ) ) |
80 |
76 79
|
sseq12d |
|- ( z = <. x , y >. -> ( ( H ` z ) C_ ( J ` z ) <-> ( x H y ) C_ ( x J y ) ) ) |
81 |
73 80
|
syl5bb |
|- ( z = <. x , y >. -> ( ( H ` z ) e. ~P ( J ` z ) <-> ( x H y ) C_ ( x J y ) ) ) |
82 |
81
|
ralxp |
|- ( A. z e. ( S X. S ) ( H ` z ) e. ~P ( J ` z ) <-> A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) |
83 |
71 82
|
bitrdi |
|- ( s = S -> ( A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) <-> A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) |
84 |
70 83
|
anbi12d |
|- ( s = S -> ( ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) <-> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) ) |
85 |
84
|
ceqsexgv |
|- ( S e. _V -> ( E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) <-> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) ) |
86 |
85
|
a1i |
|- ( ph -> ( S e. _V -> ( E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) <-> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) ) ) |
87 |
63 67 86
|
pm5.21ndd |
|- ( ph -> ( E. s ( s = S /\ ( s e. ~P T /\ A. z e. ( s X. s ) ( H ` z ) e. ~P ( J ` z ) ) ) <-> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) ) |
88 |
27 59 87
|
3bitrd |
|- ( ph -> ( H C_cat J <-> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) ) |