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Theorem seqomlem4 6669
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 4824 . . . . . . 7  |-  ( A  e.  om  ->  suc  A  e.  om )
2 fvres 5704 . . . . . . 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 6653 . . . . . . . 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 5704 . . . . . . . . . 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 5688 . . . . . . . . 9  |-  ( Q `
 suc  A )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  A
)
96, 8syl6eqr 2454 . . . . . . . 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 5704 . . . . . . . . . 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 5688 . . . . . . . . . 10  |-  ( Q `
 A )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  A )
1210, 11syl6eqr 2454 . . . . . . . . 9  |-  ( A  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  A )  =  ( Q `  A ) )
1312fveq2d 5691 . . . . . . . 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 2444 . . . . . . 7  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  A
) ) )
157seqomlem1 6666 . . . . . . . 8  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
1615fveq2d 5691 . . . . . . 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 6043 . . . . . . . 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 5701 . . . . . . . . . 10  |-  ( 2nd `  ( Q `  A
) )  e.  _V
19 suceq 4606 . . . . . . . . . . . 12  |-  ( i  =  A  ->  suc  i  =  suc  A )
20 oveq1 6047 . . . . . . . . . . . 12  |-  ( i  =  A  ->  (
i F v )  =  ( A F v ) )
2119, 20opeq12d 3952 . . . . . . . . . . 11  |-  ( i  =  A  ->  <. suc  i ,  ( i F v ) >.  =  <. suc 
A ,  ( A F v ) >.
)
22 oveq2 6048 . . . . . . . . . . . 12  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  ( A F v )  =  ( A F ( 2nd `  ( Q `
 A ) ) ) )
2322opeq2d 3951 . . . . . . . . . . 11  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  <. suc  A ,  ( A F v ) >.  =  <. suc 
A ,  ( A F ( 2nd `  ( Q `  A )
) ) >. )
24 eqid 2404 . . . . . . . . . . 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 4387 . . . . . . . . . . 11  |-  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  e.  _V
2621, 23, 24, 25ovmpt2 6168 . . . . . . . . . 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 653 . . . . . . . . 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 5704 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  ( Q `  A ) )
2928, 15eqtrd 2436 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  <. A ,  ( 2nd `  ( Q `  A )
) >. )
30 frfnom 6651 . . . . . . . . . . . . . . . . . 18  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )  |` 
om )  Fn  om
317reseq1i 5101 . . . . . . . . . . . . . . . . . . 19  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
3231fneq1i 5498 . . . . . . . . . . . . . . . . . 18  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
3330, 32mpbir 201 . . . . . . . . . . . . . . . . 17  |-  ( Q  |`  om )  Fn  om
34 fnfvelrn 5826 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Q  |`  om )  Fn  om  /\  A  e. 
om )  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3533, 34mpan 652 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3629, 35eqeltrrd 2479 . . . . . . . . . . . . . . 15  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ran  ( Q  |`  om ) )
37 df-ima 4850 . . . . . . . . . . . . . . 15  |-  ( Q
" om )  =  ran  ( Q  |`  om )
3836, 37syl6eleqr 2495 . . . . . . . . . . . . . 14  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ( Q
" om ) )
39 df-br 4173 . . . . . . . . . . . . . 14  |-  ( A ( Q " om ) ( 2nd `  ( Q `  A )
)  <->  <. A ,  ( 2nd `  ( Q `
 A ) )
>.  e.  ( Q " om ) )
4038, 39sylibr 204 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  A
( Q " om ) ( 2nd `  ( Q `  A )
) )
417seqomlem2 6667 . . . . . . . . . . . . . 14  |-  ( Q
" om )  Fn 
om
42 fnbrfvb 5726 . . . . . . . . . . . . . 14  |-  ( ( ( Q " om )  Fn  om  /\  A  e.  om )  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4341, 42mpan 652 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4440, 43mpbird 224 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  (
( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
) )
4544eqcomd 2409 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( 2nd `  ( Q `  A ) )  =  ( ( Q " om ) `  A ) )
4645oveq2d 6056 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( A F ( 2nd `  ( Q `  A )
) )  =  ( A F ( ( Q " om ) `  A ) ) )
4746opeq2d 3951 . . . . . . . . 9  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  =  <. suc 
A ,  ( A F ( ( Q
" om ) `  A ) ) >.
)
4827, 47eqtrd 2436 . . . . . . . 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 2450 . . . . . . 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 2440 . . . . . 6  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >. )
513, 50eqtrd 2436 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  = 
<. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.
)
52 fnfvelrn 5826 . . . . . 6  |-  ( ( ( Q  |`  om )  Fn  om  /\  suc  A  e.  om )  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5333, 1, 52sylancr 645 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5451, 53eqeltrrd 2479 . . . 4  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ran  ( Q  |`  om )
)
5554, 37syl6eleqr 2495 . . 3  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ( Q " om ) )
56 df-br 4173 . . 3  |-  ( suc 
A ( Q " om ) ( A F ( ( Q " om ) `  A ) )  <->  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >.  e.  ( Q
" om ) )
5755, 56sylibr 204 . 2  |-  ( A  e.  om  ->  suc  A ( Q " om ) ( A F ( ( Q " om ) `  A ) ) )
58 fnbrfvb 5726 . . 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 645 . 2  |-  ( A  e.  om  ->  (
( ( Q " om ) `  suc  A
)  =  ( A F ( ( Q
" om ) `  A ) )  <->  suc  A ( Q " om )
( A F ( ( Q " om ) `  A )
) ) )
6057, 59mpbird 224 1  |-  ( A  e.  om  ->  (
( Q " om ) `  suc  A )  =  ( A F ( ( Q " om ) `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   <.cop 3777   class class class wbr 4172    _I cid 4453   suc csuc 4543   omcom 4804   ran crn 4838    |` cres 4839   "cima 4840    Fn wfn 5408   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   2ndc2nd 6307   reccrdg 6626
This theorem is referenced by:  seqomsuc  6673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-recs 6592  df-rdg 6627
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