Step |
Hyp |
Ref |
Expression |
1 |
|
ntrcls.o |
|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) ) |
2 |
|
ntrcls.d |
|- D = ( O ` B ) |
3 |
|
ntrcls.r |
|- ( ph -> I D K ) |
4 |
|
ntrclslem0.x |
|- ( ph -> X e. B ) |
5 |
|
fveq2 |
|- ( s = t -> ( I ` s ) = ( I ` t ) ) |
6 |
5
|
eleq2d |
|- ( s = t -> ( X e. ( I ` s ) <-> X e. ( I ` t ) ) ) |
7 |
6
|
cbvrexvw |
|- ( E. s e. ~P B X e. ( I ` s ) <-> E. t e. ~P B X e. ( I ` t ) ) |
8 |
2 3
|
ntrclsrcomplex |
|- ( ph -> ( B \ s ) e. ~P B ) |
9 |
8
|
adantr |
|- ( ( ph /\ s e. ~P B ) -> ( B \ s ) e. ~P B ) |
10 |
2 3
|
ntrclsrcomplex |
|- ( ph -> ( B \ t ) e. ~P B ) |
11 |
10
|
adantr |
|- ( ( ph /\ t e. ~P B ) -> ( B \ t ) e. ~P B ) |
12 |
|
difeq2 |
|- ( s = ( B \ t ) -> ( B \ s ) = ( B \ ( B \ t ) ) ) |
13 |
12
|
adantl |
|- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> ( B \ s ) = ( B \ ( B \ t ) ) ) |
14 |
|
elpwi |
|- ( t e. ~P B -> t C_ B ) |
15 |
|
dfss4 |
|- ( t C_ B <-> ( B \ ( B \ t ) ) = t ) |
16 |
14 15
|
sylib |
|- ( t e. ~P B -> ( B \ ( B \ t ) ) = t ) |
17 |
16
|
ad2antlr |
|- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> ( B \ ( B \ t ) ) = t ) |
18 |
13 17
|
eqtr2d |
|- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> t = ( B \ s ) ) |
19 |
11 18
|
rspcedeq2vd |
|- ( ( ph /\ t e. ~P B ) -> E. s e. ~P B t = ( B \ s ) ) |
20 |
|
fveq2 |
|- ( t = ( B \ s ) -> ( I ` t ) = ( I ` ( B \ s ) ) ) |
21 |
20
|
eleq2d |
|- ( t = ( B \ s ) -> ( X e. ( I ` t ) <-> X e. ( I ` ( B \ s ) ) ) ) |
22 |
21
|
3ad2ant3 |
|- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( X e. ( I ` t ) <-> X e. ( I ` ( B \ s ) ) ) ) |
23 |
3
|
adantr |
|- ( ( ph /\ s e. ~P B ) -> I D K ) |
24 |
4
|
adantr |
|- ( ( ph /\ s e. ~P B ) -> X e. B ) |
25 |
|
simpr |
|- ( ( ph /\ s e. ~P B ) -> s e. ~P B ) |
26 |
1 2 23 24 25
|
ntrclselnel2 |
|- ( ( ph /\ s e. ~P B ) -> ( X e. ( I ` ( B \ s ) ) <-> -. X e. ( K ` s ) ) ) |
27 |
26
|
3adant3 |
|- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( X e. ( I ` ( B \ s ) ) <-> -. X e. ( K ` s ) ) ) |
28 |
22 27
|
bitrd |
|- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( X e. ( I ` t ) <-> -. X e. ( K ` s ) ) ) |
29 |
9 19 28
|
rexxfrd2 |
|- ( ph -> ( E. t e. ~P B X e. ( I ` t ) <-> E. s e. ~P B -. X e. ( K ` s ) ) ) |
30 |
7 29
|
syl5bb |
|- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> E. s e. ~P B -. X e. ( K ` s ) ) ) |