| Step |
Hyp |
Ref |
Expression |
| 1 |
|
strle1.i |
|- I e. NN |
| 2 |
|
strle1.a |
|- A = I |
| 3 |
1
|
nnrei |
|- I e. RR |
| 4 |
3
|
leidi |
|- I <_ I |
| 5 |
1 1 4
|
3pm3.2i |
|- ( I e. NN /\ I e. NN /\ I <_ I ) |
| 6 |
|
difss |
|- ( { <. A , X >. } \ { (/) } ) C_ { <. A , X >. } |
| 7 |
2 1
|
eqeltri |
|- A e. NN |
| 8 |
|
funsng |
|- ( ( A e. NN /\ X e. _V ) -> Fun { <. A , X >. } ) |
| 9 |
7 8
|
mpan |
|- ( X e. _V -> Fun { <. A , X >. } ) |
| 10 |
|
funss |
|- ( ( { <. A , X >. } \ { (/) } ) C_ { <. A , X >. } -> ( Fun { <. A , X >. } -> Fun ( { <. A , X >. } \ { (/) } ) ) ) |
| 11 |
6 9 10
|
mpsyl |
|- ( X e. _V -> Fun ( { <. A , X >. } \ { (/) } ) ) |
| 12 |
|
fun0 |
|- Fun (/) |
| 13 |
|
opprc2 |
|- ( -. X e. _V -> <. A , X >. = (/) ) |
| 14 |
13
|
sneqd |
|- ( -. X e. _V -> { <. A , X >. } = { (/) } ) |
| 15 |
14
|
difeq1d |
|- ( -. X e. _V -> ( { <. A , X >. } \ { (/) } ) = ( { (/) } \ { (/) } ) ) |
| 16 |
|
difid |
|- ( { (/) } \ { (/) } ) = (/) |
| 17 |
15 16
|
eqtrdi |
|- ( -. X e. _V -> ( { <. A , X >. } \ { (/) } ) = (/) ) |
| 18 |
17
|
funeqd |
|- ( -. X e. _V -> ( Fun ( { <. A , X >. } \ { (/) } ) <-> Fun (/) ) ) |
| 19 |
12 18
|
mpbiri |
|- ( -. X e. _V -> Fun ( { <. A , X >. } \ { (/) } ) ) |
| 20 |
11 19
|
pm2.61i |
|- Fun ( { <. A , X >. } \ { (/) } ) |
| 21 |
|
dmsnopss |
|- dom { <. A , X >. } C_ { A } |
| 22 |
2
|
sneqi |
|- { A } = { I } |
| 23 |
1
|
nnzi |
|- I e. ZZ |
| 24 |
|
fzsn |
|- ( I e. ZZ -> ( I ... I ) = { I } ) |
| 25 |
23 24
|
ax-mp |
|- ( I ... I ) = { I } |
| 26 |
22 25
|
eqtr4i |
|- { A } = ( I ... I ) |
| 27 |
21 26
|
sseqtri |
|- dom { <. A , X >. } C_ ( I ... I ) |
| 28 |
|
isstruct |
|- ( { <. A , X >. } Struct <. I , I >. <-> ( ( I e. NN /\ I e. NN /\ I <_ I ) /\ Fun ( { <. A , X >. } \ { (/) } ) /\ dom { <. A , X >. } C_ ( I ... I ) ) ) |
| 29 |
5 20 27 28
|
mpbir3an |
|- { <. A , X >. } Struct <. I , I >. |