Step |
Hyp |
Ref |
Expression |
1 |
|
cardf2 |
|- card : { x | E. y e. On y ~~ x } --> On |
2 |
|
ffun |
|- ( card : { x | E. y e. On y ~~ x } --> On -> Fun card ) |
3 |
1 2
|
ax-mp |
|- Fun card |
4 |
|
r1fnon |
|- R1 Fn On |
5 |
|
fnfun |
|- ( R1 Fn On -> Fun R1 ) |
6 |
4 5
|
ax-mp |
|- Fun R1 |
7 |
|
funco |
|- ( ( Fun card /\ Fun R1 ) -> Fun ( card o. R1 ) ) |
8 |
3 6 7
|
mp2an |
|- Fun ( card o. R1 ) |
9 |
|
funfn |
|- ( Fun ( card o. R1 ) <-> ( card o. R1 ) Fn dom ( card o. R1 ) ) |
10 |
8 9
|
mpbi |
|- ( card o. R1 ) Fn dom ( card o. R1 ) |
11 |
|
rnco |
|- ran ( card o. R1 ) = ran ( card |` ran R1 ) |
12 |
|
resss |
|- ( card |` ran R1 ) C_ card |
13 |
12
|
rnssi |
|- ran ( card |` ran R1 ) C_ ran card |
14 |
|
frn |
|- ( card : { x | E. y e. On y ~~ x } --> On -> ran card C_ On ) |
15 |
1 14
|
ax-mp |
|- ran card C_ On |
16 |
13 15
|
sstri |
|- ran ( card |` ran R1 ) C_ On |
17 |
11 16
|
eqsstri |
|- ran ( card o. R1 ) C_ On |
18 |
|
df-f |
|- ( ( card o. R1 ) : dom ( card o. R1 ) --> On <-> ( ( card o. R1 ) Fn dom ( card o. R1 ) /\ ran ( card o. R1 ) C_ On ) ) |
19 |
10 17 18
|
mpbir2an |
|- ( card o. R1 ) : dom ( card o. R1 ) --> On |
20 |
|
dmco |
|- dom ( card o. R1 ) = ( `' R1 " dom card ) |
21 |
20
|
feq2i |
|- ( ( card o. R1 ) : dom ( card o. R1 ) --> On <-> ( card o. R1 ) : ( `' R1 " dom card ) --> On ) |
22 |
19 21
|
mpbi |
|- ( card o. R1 ) : ( `' R1 " dom card ) --> On |
23 |
|
elpreima |
|- ( R1 Fn On -> ( x e. ( `' R1 " dom card ) <-> ( x e. On /\ ( R1 ` x ) e. dom card ) ) ) |
24 |
4 23
|
ax-mp |
|- ( x e. ( `' R1 " dom card ) <-> ( x e. On /\ ( R1 ` x ) e. dom card ) ) |
25 |
24
|
simplbi |
|- ( x e. ( `' R1 " dom card ) -> x e. On ) |
26 |
|
onelon |
|- ( ( x e. On /\ y e. x ) -> y e. On ) |
27 |
25 26
|
sylan |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> y e. On ) |
28 |
24
|
simprbi |
|- ( x e. ( `' R1 " dom card ) -> ( R1 ` x ) e. dom card ) |
29 |
28
|
adantr |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` x ) e. dom card ) |
30 |
|
r1ord2 |
|- ( x e. On -> ( y e. x -> ( R1 ` y ) C_ ( R1 ` x ) ) ) |
31 |
30
|
imp |
|- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) C_ ( R1 ` x ) ) |
32 |
25 31
|
sylan |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` y ) C_ ( R1 ` x ) ) |
33 |
|
ssnum |
|- ( ( ( R1 ` x ) e. dom card /\ ( R1 ` y ) C_ ( R1 ` x ) ) -> ( R1 ` y ) e. dom card ) |
34 |
29 32 33
|
syl2anc |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` y ) e. dom card ) |
35 |
|
elpreima |
|- ( R1 Fn On -> ( y e. ( `' R1 " dom card ) <-> ( y e. On /\ ( R1 ` y ) e. dom card ) ) ) |
36 |
4 35
|
ax-mp |
|- ( y e. ( `' R1 " dom card ) <-> ( y e. On /\ ( R1 ` y ) e. dom card ) ) |
37 |
27 34 36
|
sylanbrc |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> y e. ( `' R1 " dom card ) ) |
38 |
37
|
rgen2 |
|- A. x e. ( `' R1 " dom card ) A. y e. x y e. ( `' R1 " dom card ) |
39 |
|
dftr5 |
|- ( Tr ( `' R1 " dom card ) <-> A. x e. ( `' R1 " dom card ) A. y e. x y e. ( `' R1 " dom card ) ) |
40 |
38 39
|
mpbir |
|- Tr ( `' R1 " dom card ) |
41 |
|
cnvimass |
|- ( `' R1 " dom card ) C_ dom R1 |
42 |
|
dffn2 |
|- ( R1 Fn On <-> R1 : On --> _V ) |
43 |
4 42
|
mpbi |
|- R1 : On --> _V |
44 |
43
|
fdmi |
|- dom R1 = On |
45 |
41 44
|
sseqtri |
|- ( `' R1 " dom card ) C_ On |
46 |
|
epweon |
|- _E We On |
47 |
|
wess |
|- ( ( `' R1 " dom card ) C_ On -> ( _E We On -> _E We ( `' R1 " dom card ) ) ) |
48 |
45 46 47
|
mp2 |
|- _E We ( `' R1 " dom card ) |
49 |
|
df-ord |
|- ( Ord ( `' R1 " dom card ) <-> ( Tr ( `' R1 " dom card ) /\ _E We ( `' R1 " dom card ) ) ) |
50 |
40 48 49
|
mpbir2an |
|- Ord ( `' R1 " dom card ) |
51 |
|
r1sdom |
|- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) ~< ( R1 ` x ) ) |
52 |
25 51
|
sylan |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` y ) ~< ( R1 ` x ) ) |
53 |
|
cardsdom2 |
|- ( ( ( R1 ` y ) e. dom card /\ ( R1 ` x ) e. dom card ) -> ( ( card ` ( R1 ` y ) ) e. ( card ` ( R1 ` x ) ) <-> ( R1 ` y ) ~< ( R1 ` x ) ) ) |
54 |
34 29 53
|
syl2anc |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card ` ( R1 ` y ) ) e. ( card ` ( R1 ` x ) ) <-> ( R1 ` y ) ~< ( R1 ` x ) ) ) |
55 |
52 54
|
mpbird |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( card ` ( R1 ` y ) ) e. ( card ` ( R1 ` x ) ) ) |
56 |
|
fvco2 |
|- ( ( R1 Fn On /\ y e. On ) -> ( ( card o. R1 ) ` y ) = ( card ` ( R1 ` y ) ) ) |
57 |
4 27 56
|
sylancr |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` y ) = ( card ` ( R1 ` y ) ) ) |
58 |
25
|
adantr |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> x e. On ) |
59 |
|
fvco2 |
|- ( ( R1 Fn On /\ x e. On ) -> ( ( card o. R1 ) ` x ) = ( card ` ( R1 ` x ) ) ) |
60 |
4 58 59
|
sylancr |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` x ) = ( card ` ( R1 ` x ) ) ) |
61 |
55 57 60
|
3eltr4d |
|- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) ) |
62 |
61
|
ex |
|- ( x e. ( `' R1 " dom card ) -> ( y e. x -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) ) ) |
63 |
62
|
adantl |
|- ( ( y e. ( `' R1 " dom card ) /\ x e. ( `' R1 " dom card ) ) -> ( y e. x -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) ) ) |
64 |
22 50 63 20
|
issmo |
|- Smo ( card o. R1 ) |