Step |
Hyp |
Ref |
Expression |
1 |
|
seqomlem.a |
|- Q = rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |
2 |
|
peano2 |
|- ( A e. _om -> suc A e. _om ) |
3 |
2
|
fvresd |
|- ( A e. _om -> ( ( Q |` _om ) ` suc A ) = ( Q ` suc A ) ) |
4 |
|
frsuc |
|- ( A e. _om -> ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` suc A ) = ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` A ) ) ) |
5 |
2
|
fvresd |
|- ( A e. _om -> ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` suc A ) = ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc A ) ) |
6 |
1
|
fveq1i |
|- ( Q ` suc A ) = ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` suc A ) |
7 |
5 6
|
eqtr4di |
|- ( A e. _om -> ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` suc A ) = ( Q ` suc A ) ) |
8 |
|
fvres |
|- ( A e. _om -> ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` A ) = ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` A ) ) |
9 |
1
|
fveq1i |
|- ( Q ` A ) = ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) ` A ) |
10 |
8 9
|
eqtr4di |
|- ( A e. _om -> ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` A ) = ( Q ` A ) ) |
11 |
10
|
fveq2d |
|- ( A e. _om -> ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) ` A ) ) = ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` A ) ) ) |
12 |
4 7 11
|
3eqtr3d |
|- ( A e. _om -> ( Q ` suc A ) = ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` A ) ) ) |
13 |
1
|
seqomlem1 |
|- ( A e. _om -> ( Q ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. ) |
14 |
13
|
fveq2d |
|- ( A e. _om -> ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` ( Q ` A ) ) = ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. A , ( 2nd ` ( Q ` A ) ) >. ) ) |
15 |
|
df-ov |
|- ( A ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. A , ( 2nd ` ( Q ` A ) ) >. ) |
16 |
|
fvex |
|- ( 2nd ` ( Q ` A ) ) e. _V |
17 |
|
suceq |
|- ( i = A -> suc i = suc A ) |
18 |
|
oveq1 |
|- ( i = A -> ( i F v ) = ( A F v ) ) |
19 |
17 18
|
opeq12d |
|- ( i = A -> <. suc i , ( i F v ) >. = <. suc A , ( A F v ) >. ) |
20 |
|
oveq2 |
|- ( v = ( 2nd ` ( Q ` A ) ) -> ( A F v ) = ( A F ( 2nd ` ( Q ` A ) ) ) ) |
21 |
20
|
opeq2d |
|- ( v = ( 2nd ` ( Q ` A ) ) -> <. suc A , ( A F v ) >. = <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. ) |
22 |
|
eqid |
|- ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) = ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) |
23 |
|
opex |
|- <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. e. _V |
24 |
19 21 22 23
|
ovmpo |
|- ( ( A e. _om /\ ( 2nd ` ( Q ` A ) ) e. _V ) -> ( A ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. ) |
25 |
16 24
|
mpan2 |
|- ( A e. _om -> ( A ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. ) |
26 |
|
fvres |
|- ( A e. _om -> ( ( Q |` _om ) ` A ) = ( Q ` A ) ) |
27 |
26 13
|
eqtrd |
|- ( A e. _om -> ( ( Q |` _om ) ` A ) = <. A , ( 2nd ` ( Q ` A ) ) >. ) |
28 |
|
frfnom |
|- ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) Fn _om |
29 |
1
|
reseq1i |
|- ( Q |` _om ) = ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) |
30 |
29
|
fneq1i |
|- ( ( Q |` _om ) Fn _om <-> ( rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) |` _om ) Fn _om ) |
31 |
28 30
|
mpbir |
|- ( Q |` _om ) Fn _om |
32 |
|
fnfvelrn |
|- ( ( ( Q |` _om ) Fn _om /\ A e. _om ) -> ( ( Q |` _om ) ` A ) e. ran ( Q |` _om ) ) |
33 |
31 32
|
mpan |
|- ( A e. _om -> ( ( Q |` _om ) ` A ) e. ran ( Q |` _om ) ) |
34 |
27 33
|
eqeltrrd |
|- ( A e. _om -> <. A , ( 2nd ` ( Q ` A ) ) >. e. ran ( Q |` _om ) ) |
35 |
|
df-ima |
|- ( Q " _om ) = ran ( Q |` _om ) |
36 |
34 35
|
eleqtrrdi |
|- ( A e. _om -> <. A , ( 2nd ` ( Q ` A ) ) >. e. ( Q " _om ) ) |
37 |
|
df-br |
|- ( A ( Q " _om ) ( 2nd ` ( Q ` A ) ) <-> <. A , ( 2nd ` ( Q ` A ) ) >. e. ( Q " _om ) ) |
38 |
36 37
|
sylibr |
|- ( A e. _om -> A ( Q " _om ) ( 2nd ` ( Q ` A ) ) ) |
39 |
1
|
seqomlem2 |
|- ( Q " _om ) Fn _om |
40 |
|
fnbrfvb |
|- ( ( ( Q " _om ) Fn _om /\ A e. _om ) -> ( ( ( Q " _om ) ` A ) = ( 2nd ` ( Q ` A ) ) <-> A ( Q " _om ) ( 2nd ` ( Q ` A ) ) ) ) |
41 |
39 40
|
mpan |
|- ( A e. _om -> ( ( ( Q " _om ) ` A ) = ( 2nd ` ( Q ` A ) ) <-> A ( Q " _om ) ( 2nd ` ( Q ` A ) ) ) ) |
42 |
38 41
|
mpbird |
|- ( A e. _om -> ( ( Q " _om ) ` A ) = ( 2nd ` ( Q ` A ) ) ) |
43 |
42
|
eqcomd |
|- ( A e. _om -> ( 2nd ` ( Q ` A ) ) = ( ( Q " _om ) ` A ) ) |
44 |
43
|
oveq2d |
|- ( A e. _om -> ( A F ( 2nd ` ( Q ` A ) ) ) = ( A F ( ( Q " _om ) ` A ) ) ) |
45 |
44
|
opeq2d |
|- ( A e. _om -> <. suc A , ( A F ( 2nd ` ( Q ` A ) ) ) >. = <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. ) |
46 |
25 45
|
eqtrd |
|- ( A e. _om -> ( A ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ( 2nd ` ( Q ` A ) ) ) = <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. ) |
47 |
15 46
|
eqtr3id |
|- ( A e. _om -> ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) ` <. A , ( 2nd ` ( Q ` A ) ) >. ) = <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. ) |
48 |
12 14 47
|
3eqtrd |
|- ( A e. _om -> ( Q ` suc A ) = <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. ) |
49 |
3 48
|
eqtrd |
|- ( A e. _om -> ( ( Q |` _om ) ` suc A ) = <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. ) |
50 |
|
fnfvelrn |
|- ( ( ( Q |` _om ) Fn _om /\ suc A e. _om ) -> ( ( Q |` _om ) ` suc A ) e. ran ( Q |` _om ) ) |
51 |
31 2 50
|
sylancr |
|- ( A e. _om -> ( ( Q |` _om ) ` suc A ) e. ran ( Q |` _om ) ) |
52 |
49 51
|
eqeltrrd |
|- ( A e. _om -> <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. e. ran ( Q |` _om ) ) |
53 |
52 35
|
eleqtrrdi |
|- ( A e. _om -> <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. e. ( Q " _om ) ) |
54 |
|
df-br |
|- ( suc A ( Q " _om ) ( A F ( ( Q " _om ) ` A ) ) <-> <. suc A , ( A F ( ( Q " _om ) ` A ) ) >. e. ( Q " _om ) ) |
55 |
53 54
|
sylibr |
|- ( A e. _om -> suc A ( Q " _om ) ( A F ( ( Q " _om ) ` A ) ) ) |
56 |
|
fnbrfvb |
|- ( ( ( Q " _om ) Fn _om /\ suc A e. _om ) -> ( ( ( Q " _om ) ` suc A ) = ( A F ( ( Q " _om ) ` A ) ) <-> suc A ( Q " _om ) ( A F ( ( Q " _om ) ` A ) ) ) ) |
57 |
39 2 56
|
sylancr |
|- ( A e. _om -> ( ( ( Q " _om ) ` suc A ) = ( A F ( ( Q " _om ) ` A ) ) <-> suc A ( Q " _om ) ( A F ( ( Q " _om ) ` A ) ) ) ) |
58 |
55 57
|
mpbird |
|- ( A e. _om -> ( ( Q " _om ) ` suc A ) = ( A F ( ( Q " _om ) ` A ) ) ) |