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Theorem seqomlem4 7118
Description: Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
Hypothesis
Ref Expression
seqomlem.a  |-  Q  =  rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)
Assertion
Ref Expression
seqomlem4  |-  ( A  e.  om  ->  (
( Q " om ) `  suc  A )  =  ( A F ( ( Q " om ) `  A ) ) )
Distinct variable groups:    Q, i,
v    A, i, v    i, F, v
Allowed substitution hints:    I( v, i)

Proof of Theorem seqomlem4
StepHypRef Expression
1 peano2 6704 . . . . . . 7  |-  ( A  e.  om  ->  suc  A  e.  om )
2 fvres 5880 . . . . . . 7  |-  ( suc 
A  e.  om  ->  ( ( Q  |`  om ) `  suc  A )  =  ( Q `  suc  A ) )
31, 2syl 16 . . . . . 6  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  =  ( Q `  suc  A ) )
4 frsuc 7102 . . . . . . . 8  |-  ( 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 fvres 5880 . . . . . . . . . 10  |-  ( suc 
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
) )
61, 5syl 16 . . . . . . . . 9  |-  ( 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
) )
7 seqomlem.a . . . . . . . . . 10  |-  Q  =  rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)
87fveq1i 5867 . . . . . . . . 9  |-  ( Q `
 suc  A )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  A
)
96, 8syl6eqr 2526 . . . . . . . 8  |-  ( A  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  suc  A )  =  ( Q `  suc  A ) )
10 fvres 5880 . . . . . . . . . 10  |-  ( 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 )
)
117fveq1i 5867 . . . . . . . . . 10  |-  ( Q `
 A )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  A )
1210, 11syl6eqr 2526 . . . . . . . . 9  |-  ( A  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  A )  =  ( Q `  A ) )
1312fveq2d 5870 . . . . . . . 8  |-  ( 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 )
) )
144, 9, 133eqtr3d 2516 . . . . . . 7  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  A
) ) )
157seqomlem1 7115 . . . . . . . 8  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
1615fveq2d 5870 . . . . . . 7  |-  ( 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 ) )
>. ) )
17 df-ov 6287 . . . . . . . 8  |-  ( 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 )
) >. )
18 fvex 5876 . . . . . . . . . 10  |-  ( 2nd `  ( Q `  A
) )  e.  _V
19 suceq 4943 . . . . . . . . . . . 12  |-  ( i  =  A  ->  suc  i  =  suc  A )
20 oveq1 6291 . . . . . . . . . . . 12  |-  ( i  =  A  ->  (
i F v )  =  ( A F v ) )
2119, 20opeq12d 4221 . . . . . . . . . . 11  |-  ( i  =  A  ->  <. suc  i ,  ( i F v ) >.  =  <. suc 
A ,  ( A F v ) >.
)
22 oveq2 6292 . . . . . . . . . . . 12  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  ( A F v )  =  ( A F ( 2nd `  ( Q `
 A ) ) ) )
2322opeq2d 4220 . . . . . . . . . . 11  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  <. suc  A ,  ( A F v ) >.  =  <. suc 
A ,  ( A F ( 2nd `  ( Q `  A )
) ) >. )
24 eqid 2467 . . . . . . . . . . 11  |-  ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. )  =  ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. )
25 opex 4711 . . . . . . . . . . 11  |-  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  e.  _V
2621, 23, 24, 25ovmpt2 6422 . . . . . . . . . 10  |-  ( ( 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 )
) ) >. )
2718, 26mpan2 671 . . . . . . . . 9  |-  ( 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
) ) ) >.
)
28 fvres 5880 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  ( Q `  A ) )
2928, 15eqtrd 2508 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  <. A ,  ( 2nd `  ( Q `  A )
) >. )
30 frfnom 7100 . . . . . . . . . . . . . . . . . 18  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )  |` 
om )  Fn  om
317reseq1i 5269 . . . . . . . . . . . . . . . . . . 19  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
3231fneq1i 5675 . . . . . . . . . . . . . . . . . 18  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
3330, 32mpbir 209 . . . . . . . . . . . . . . . . 17  |-  ( Q  |`  om )  Fn  om
34 fnfvelrn 6018 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Q  |`  om )  Fn  om  /\  A  e. 
om )  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3533, 34mpan 670 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3629, 35eqeltrrd 2556 . . . . . . . . . . . . . . 15  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ran  ( Q  |`  om ) )
37 df-ima 5012 . . . . . . . . . . . . . . 15  |-  ( Q
" om )  =  ran  ( Q  |`  om )
3836, 37syl6eleqr 2566 . . . . . . . . . . . . . 14  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ( Q
" om ) )
39 df-br 4448 . . . . . . . . . . . . . 14  |-  ( A ( Q " om ) ( 2nd `  ( Q `  A )
)  <->  <. A ,  ( 2nd `  ( Q `
 A ) )
>.  e.  ( Q " om ) )
4038, 39sylibr 212 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  A
( Q " om ) ( 2nd `  ( Q `  A )
) )
417seqomlem2 7116 . . . . . . . . . . . . . 14  |-  ( Q
" om )  Fn 
om
42 fnbrfvb 5908 . . . . . . . . . . . . . 14  |-  ( ( ( Q " om )  Fn  om  /\  A  e.  om )  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4341, 42mpan 670 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4440, 43mpbird 232 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  (
( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
) )
4544eqcomd 2475 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( 2nd `  ( Q `  A ) )  =  ( ( Q " om ) `  A ) )
4645oveq2d 6300 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( A F ( 2nd `  ( Q `  A )
) )  =  ( A F ( ( Q " om ) `  A ) ) )
4746opeq2d 4220 . . . . . . . . 9  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  =  <. suc 
A ,  ( A F ( ( Q
" om ) `  A ) ) >.
)
4827, 47eqtrd 2508 . . . . . . . 8  |-  ( 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 ) ) >.
)
4917, 48syl5eqr 2522 . . . . . . 7  |-  ( 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 ) ) >.
)
5014, 16, 493eqtrd 2512 . . . . . 6  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >. )
513, 50eqtrd 2508 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  = 
<. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.
)
52 fnfvelrn 6018 . . . . . 6  |-  ( ( ( Q  |`  om )  Fn  om  /\  suc  A  e.  om )  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5333, 1, 52sylancr 663 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5451, 53eqeltrrd 2556 . . . 4  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ran  ( Q  |`  om )
)
5554, 37syl6eleqr 2566 . . 3  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ( Q " om ) )
56 df-br 4448 . . 3  |-  ( suc 
A ( Q " om ) ( A F ( ( Q " om ) `  A ) )  <->  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >.  e.  ( Q
" om ) )
5755, 56sylibr 212 . 2  |-  ( A  e.  om  ->  suc  A ( Q " om ) ( A F ( ( Q " om ) `  A ) ) )
58 fnbrfvb 5908 . . 3  |-  ( ( ( 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 )
) ) )
5941, 1, 58sylancr 663 . 2  |-  ( A  e.  om  ->  (
( ( Q " om ) `  suc  A
)  =  ( A F ( ( Q
" om ) `  A ) )  <->  suc  A ( Q " om )
( A F ( ( Q " om ) `  A )
) ) )
6057, 59mpbird 232 1  |-  ( A  e.  om  ->  (
( Q " om ) `  suc  A )  =  ( A F ( ( Q " om ) `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   <.cop 4033   class class class wbr 4447    _I cid 4790   suc csuc 4880   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5583   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   omcom 6684   2ndc2nd 6783   reccrdg 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-2nd 6785  df-recs 7042  df-rdg 7076
This theorem is referenced by:  seqomsuc  7122
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