Step |
Hyp |
Ref |
Expression |
1 |
|
dmres |
|- dom ( A |` suc X ) = ( suc X i^i dom A ) |
2 |
|
simp11 |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A A e. No ) |
3 |
|
nodmord |
|- ( A e. No -> Ord dom A ) |
4 |
2 3
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A Ord dom A ) |
5 |
|
ndmfv |
|- ( -. X e. dom A -> ( A ` X ) = (/) ) |
6 |
|
1n0 |
|- 1o =/= (/) |
7 |
6
|
necomi |
|- (/) =/= 1o |
8 |
|
neeq1 |
|- ( ( A ` X ) = (/) -> ( ( A ` X ) =/= 1o <-> (/) =/= 1o ) ) |
9 |
7 8
|
mpbiri |
|- ( ( A ` X ) = (/) -> ( A ` X ) =/= 1o ) |
10 |
9
|
neneqd |
|- ( ( A ` X ) = (/) -> -. ( A ` X ) = 1o ) |
11 |
5 10
|
syl |
|- ( -. X e. dom A -> -. ( A ` X ) = 1o ) |
12 |
11
|
con4i |
|- ( ( A ` X ) = 1o -> X e. dom A ) |
13 |
12
|
adantl |
|- ( ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) -> X e. dom A ) |
14 |
13
|
3ad2ant2 |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A X e. dom A ) |
15 |
|
ordsucss |
|- ( Ord dom A -> ( X e. dom A -> suc X C_ dom A ) ) |
16 |
4 14 15
|
sylc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A suc X C_ dom A ) |
17 |
|
df-ss |
|- ( suc X C_ dom A <-> ( suc X i^i dom A ) = suc X ) |
18 |
16 17
|
sylib |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( suc X i^i dom A ) = suc X ) |
19 |
1 18
|
syl5eq |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A dom ( A |` suc X ) = suc X ) |
20 |
|
dmres |
|- dom ( B |` suc X ) = ( suc X i^i dom B ) |
21 |
|
simp12 |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A B e. No ) |
22 |
|
nodmord |
|- ( B e. No -> Ord dom B ) |
23 |
21 22
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A Ord dom B ) |
24 |
|
nogesgn1o |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( B ` X ) = 1o ) |
25 |
|
ndmfv |
|- ( -. X e. dom B -> ( B ` X ) = (/) ) |
26 |
|
neeq1 |
|- ( ( B ` X ) = (/) -> ( ( B ` X ) =/= 1o <-> (/) =/= 1o ) ) |
27 |
7 26
|
mpbiri |
|- ( ( B ` X ) = (/) -> ( B ` X ) =/= 1o ) |
28 |
27
|
neneqd |
|- ( ( B ` X ) = (/) -> -. ( B ` X ) = 1o ) |
29 |
25 28
|
syl |
|- ( -. X e. dom B -> -. ( B ` X ) = 1o ) |
30 |
29
|
con4i |
|- ( ( B ` X ) = 1o -> X e. dom B ) |
31 |
24 30
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A X e. dom B ) |
32 |
|
ordsucss |
|- ( Ord dom B -> ( X e. dom B -> suc X C_ dom B ) ) |
33 |
23 31 32
|
sylc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A suc X C_ dom B ) |
34 |
|
df-ss |
|- ( suc X C_ dom B <-> ( suc X i^i dom B ) = suc X ) |
35 |
33 34
|
sylib |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( suc X i^i dom B ) = suc X ) |
36 |
20 35
|
syl5eq |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A dom ( B |` suc X ) = suc X ) |
37 |
19 36
|
eqtr4d |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A dom ( A |` suc X ) = dom ( B |` suc X ) ) |
38 |
19
|
eleq2d |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( x e. dom ( A |` suc X ) <-> x e. suc X ) ) |
39 |
|
vex |
|- x e. _V |
40 |
39
|
elsuc |
|- ( x e. suc X <-> ( x e. X \/ x = X ) ) |
41 |
|
simpl2l |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( A |` X ) = ( B |` X ) ) |
42 |
41
|
fveq1d |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( A |` X ) ` x ) = ( ( B |` X ) ` x ) ) |
43 |
|
simpr |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A x e. X ) |
44 |
43
|
fvresd |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( A |` X ) ` x ) = ( A ` x ) ) |
45 |
43
|
fvresd |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( B |` X ) ` x ) = ( B ` x ) ) |
46 |
42 44 45
|
3eqtr3d |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( A ` x ) = ( B ` x ) ) |
47 |
46
|
ex |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( x e. X -> ( A ` x ) = ( B ` x ) ) ) |
48 |
|
simp2r |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( A ` X ) = 1o ) |
49 |
48 24
|
eqtr4d |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( A ` X ) = ( B ` X ) ) |
50 |
|
fveq2 |
|- ( x = X -> ( A ` x ) = ( A ` X ) ) |
51 |
|
fveq2 |
|- ( x = X -> ( B ` x ) = ( B ` X ) ) |
52 |
50 51
|
eqeq12d |
|- ( x = X -> ( ( A ` x ) = ( B ` x ) <-> ( A ` X ) = ( B ` X ) ) ) |
53 |
49 52
|
syl5ibrcom |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( x = X -> ( A ` x ) = ( B ` x ) ) ) |
54 |
47 53
|
jaod |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( x e. X \/ x = X ) -> ( A ` x ) = ( B ` x ) ) ) |
55 |
40 54
|
syl5bi |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( x e. suc X -> ( A ` x ) = ( B ` x ) ) ) |
56 |
55
|
imp |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( A ` x ) = ( B ` x ) ) |
57 |
|
simpr |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A x e. suc X ) |
58 |
57
|
fvresd |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( A |` suc X ) ` x ) = ( A ` x ) ) |
59 |
57
|
fvresd |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( B |` suc X ) ` x ) = ( B ` x ) ) |
60 |
56 58 59
|
3eqtr4d |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) |
61 |
60
|
ex |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( x e. suc X -> ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) ) |
62 |
38 61
|
sylbid |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( x e. dom ( A |` suc X ) -> ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) ) |
63 |
62
|
ralrimiv |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A A. x e. dom ( A |` suc X ) ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) |
64 |
|
nofun |
|- ( A e. No -> Fun A ) |
65 |
2 64
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A Fun A ) |
66 |
65
|
funresd |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A Fun ( A |` suc X ) ) |
67 |
|
nofun |
|- ( B e. No -> Fun B ) |
68 |
21 67
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A Fun B ) |
69 |
68
|
funresd |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A Fun ( B |` suc X ) ) |
70 |
|
eqfunfv |
|- ( ( Fun ( A |` suc X ) /\ Fun ( B |` suc X ) ) -> ( ( A |` suc X ) = ( B |` suc X ) <-> ( dom ( A |` suc X ) = dom ( B |` suc X ) /\ A. x e. dom ( A |` suc X ) ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) ) ) |
71 |
66 69 70
|
syl2anc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( ( A |` suc X ) = ( B |` suc X ) <-> ( dom ( A |` suc X ) = dom ( B |` suc X ) /\ A. x e. dom ( A |` suc X ) ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) ) ) |
72 |
37 63 71
|
mpbir2and |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A ( A |` suc X ) = ( B |` suc X ) ) |