Step |
Hyp |
Ref |
Expression |
1 |
|
nfcv |
|- F/_ n if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) |
2 |
|
nfcv |
|- F/_ m if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) ) |
3 |
|
fveqeq2 |
|- ( m = n -> ( ( F ` m ) = (/) <-> ( F ` n ) = (/) ) ) |
4 |
|
fveq2 |
|- ( m = n -> ( F ` m ) = ( F ` n ) ) |
5 |
3 4
|
ifbieq2d |
|- ( m = n -> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) = if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) ) ) |
6 |
1 2 5
|
cbvmpt |
|- ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) = ( n e. _om |-> if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) ) ) |
7 |
|
nfcv |
|- F/_ n ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) |
8 |
|
nfcv |
|- F/_ m { n } |
9 |
|
nffvmpt1 |
|- F/_ m ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) |
10 |
8 9
|
nfxp |
|- F/_ m ( { n } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) ) |
11 |
|
sneq |
|- ( m = n -> { m } = { n } ) |
12 |
|
fveq2 |
|- ( m = n -> ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) = ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) ) |
13 |
11 12
|
xpeq12d |
|- ( m = n -> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) = ( { n } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) ) ) |
14 |
7 10 13
|
cbvmpt |
|- ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) = ( n e. _om |-> ( { n } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) ) ) |
15 |
|
nfcv |
|- F/_ n ( 2nd ` ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) ) |
16 |
|
nfcv |
|- F/_ m 2nd |
17 |
|
nfcv |
|- F/_ m f |
18 |
|
nffvmpt1 |
|- F/_ m ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) |
19 |
17 18
|
nffv |
|- F/_ m ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) |
20 |
16 19
|
nffv |
|- F/_ m ( 2nd ` ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) ) |
21 |
|
2fveq3 |
|- ( m = n -> ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) = ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) ) |
22 |
21
|
fveq2d |
|- ( m = n -> ( 2nd ` ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) ) = ( 2nd ` ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) ) ) |
23 |
15 20 22
|
cbvmpt |
|- ( m e. _om |-> ( 2nd ` ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) ) ) = ( n e. _om |-> ( 2nd ` ( f ` ( ( m e. _om |-> ( { m } X. ( ( m e. _om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) ) ) |
24 |
6 14 23
|
axcc2lem |
|- E. g ( g Fn _om /\ A. n e. _om ( ( F ` n ) =/= (/) -> ( g ` n ) e. ( F ` n ) ) ) |