Step |
Hyp |
Ref |
Expression |
1 |
|
inf3lem.1 |
|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) |
2 |
|
inf3lem.2 |
|- F = ( rec ( G , (/) ) |` _om ) |
3 |
|
inf3lem.3 |
|- A e. _V |
4 |
|
inf3lem.4 |
|- B e. _V |
5 |
|
vex |
|- u e. _V |
6 |
|
vex |
|- v e. _V |
7 |
1 2 5 6
|
inf3lem5 |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. _om /\ v e. u ) -> ( F ` v ) C. ( F ` u ) ) ) |
8 |
|
dfpss2 |
|- ( ( F ` v ) C. ( F ` u ) <-> ( ( F ` v ) C_ ( F ` u ) /\ -. ( F ` v ) = ( F ` u ) ) ) |
9 |
8
|
simprbi |
|- ( ( F ` v ) C. ( F ` u ) -> -. ( F ` v ) = ( F ` u ) ) |
10 |
7 9
|
syl6 |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( u e. _om /\ v e. u ) -> -. ( F ` v ) = ( F ` u ) ) ) |
11 |
10
|
expdimp |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ u e. _om ) -> ( v e. u -> -. ( F ` v ) = ( F ` u ) ) ) |
12 |
11
|
adantrl |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. _om /\ u e. _om ) ) -> ( v e. u -> -. ( F ` v ) = ( F ` u ) ) ) |
13 |
1 2 6 5
|
inf3lem5 |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. _om /\ u e. v ) -> ( F ` u ) C. ( F ` v ) ) ) |
14 |
|
dfpss2 |
|- ( ( F ` u ) C. ( F ` v ) <-> ( ( F ` u ) C_ ( F ` v ) /\ -. ( F ` u ) = ( F ` v ) ) ) |
15 |
14
|
simprbi |
|- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` u ) = ( F ` v ) ) |
16 |
|
eqcom |
|- ( ( F ` u ) = ( F ` v ) <-> ( F ` v ) = ( F ` u ) ) |
17 |
15 16
|
sylnib |
|- ( ( F ` u ) C. ( F ` v ) -> -. ( F ` v ) = ( F ` u ) ) |
18 |
13 17
|
syl6 |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( ( v e. _om /\ u e. v ) -> -. ( F ` v ) = ( F ` u ) ) ) |
19 |
18
|
expdimp |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ v e. _om ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) ) |
20 |
19
|
adantrr |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. _om /\ u e. _om ) ) -> ( u e. v -> -. ( F ` v ) = ( F ` u ) ) ) |
21 |
12 20
|
jaod |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. _om /\ u e. _om ) ) -> ( ( v e. u \/ u e. v ) -> -. ( F ` v ) = ( F ` u ) ) ) |
22 |
21
|
con2d |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. _om /\ u e. _om ) ) -> ( ( F ` v ) = ( F ` u ) -> -. ( v e. u \/ u e. v ) ) ) |
23 |
|
nnord |
|- ( v e. _om -> Ord v ) |
24 |
|
nnord |
|- ( u e. _om -> Ord u ) |
25 |
|
ordtri3 |
|- ( ( Ord v /\ Ord u ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) ) |
26 |
23 24 25
|
syl2an |
|- ( ( v e. _om /\ u e. _om ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) ) |
27 |
26
|
adantl |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. _om /\ u e. _om ) ) -> ( v = u <-> -. ( v e. u \/ u e. v ) ) ) |
28 |
22 27
|
sylibrd |
|- ( ( ( x =/= (/) /\ x C_ U. x ) /\ ( v e. _om /\ u e. _om ) ) -> ( ( F ` v ) = ( F ` u ) -> v = u ) ) |
29 |
28
|
ralrimivva |
|- ( ( x =/= (/) /\ x C_ U. x ) -> A. v e. _om A. u e. _om ( ( F ` v ) = ( F ` u ) -> v = u ) ) |
30 |
|
frfnom |
|- ( rec ( G , (/) ) |` _om ) Fn _om |
31 |
|
fneq1 |
|- ( F = ( rec ( G , (/) ) |` _om ) -> ( F Fn _om <-> ( rec ( G , (/) ) |` _om ) Fn _om ) ) |
32 |
30 31
|
mpbiri |
|- ( F = ( rec ( G , (/) ) |` _om ) -> F Fn _om ) |
33 |
|
fvelrnb |
|- ( F Fn _om -> ( u e. ran F <-> E. v e. _om ( F ` v ) = u ) ) |
34 |
1 2 6 4
|
inf3lemd |
|- ( v e. _om -> ( F ` v ) C_ x ) |
35 |
|
fvex |
|- ( F ` v ) e. _V |
36 |
35
|
elpw |
|- ( ( F ` v ) e. ~P x <-> ( F ` v ) C_ x ) |
37 |
34 36
|
sylibr |
|- ( v e. _om -> ( F ` v ) e. ~P x ) |
38 |
|
eleq1 |
|- ( ( F ` v ) = u -> ( ( F ` v ) e. ~P x <-> u e. ~P x ) ) |
39 |
37 38
|
syl5ibcom |
|- ( v e. _om -> ( ( F ` v ) = u -> u e. ~P x ) ) |
40 |
39
|
rexlimiv |
|- ( E. v e. _om ( F ` v ) = u -> u e. ~P x ) |
41 |
33 40
|
syl6bi |
|- ( F Fn _om -> ( u e. ran F -> u e. ~P x ) ) |
42 |
41
|
ssrdv |
|- ( F Fn _om -> ran F C_ ~P x ) |
43 |
42
|
ancli |
|- ( F Fn _om -> ( F Fn _om /\ ran F C_ ~P x ) ) |
44 |
2 32 43
|
mp2b |
|- ( F Fn _om /\ ran F C_ ~P x ) |
45 |
|
df-f |
|- ( F : _om --> ~P x <-> ( F Fn _om /\ ran F C_ ~P x ) ) |
46 |
44 45
|
mpbir |
|- F : _om --> ~P x |
47 |
29 46
|
jctil |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( F : _om --> ~P x /\ A. v e. _om A. u e. _om ( ( F ` v ) = ( F ` u ) -> v = u ) ) ) |
48 |
|
dff13 |
|- ( F : _om -1-1-> ~P x <-> ( F : _om --> ~P x /\ A. v e. _om A. u e. _om ( ( F ` v ) = ( F ` u ) -> v = u ) ) ) |
49 |
47 48
|
sylibr |
|- ( ( x =/= (/) /\ x C_ U. x ) -> F : _om -1-1-> ~P x ) |