Metamath Proof Explorer


Theorem seqomlem4

Description: Lemma for seqom . (Contributed by Stefan O'Rear, 1-Nov-2014) (Revised by Mario Carneiro, 23-Jun-2015)

Ref Expression
Hypothesis seqomlem.a
|- Q = rec ( ( i e. _om , v e. _V |-> <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. )
Assertion seqomlem4
|- ( A e. _om -> ( ( Q " _om ) ` suc A ) = ( A F ( ( Q " _om ) ` A ) ) )

Proof

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 ) ) )