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Theorem seqomlem4 6465
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 4676 . . . . . . 7  |-  ( A  e.  om  ->  suc  A  e.  om )
2 fvres 5542 . . . . . . 7  |-  ( suc 
A  e.  om  ->  ( ( Q  |`  om ) `  suc  A )  =  ( Q `  suc  A ) )
31, 2syl 15 . . . . . 6  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  =  ( Q `  suc  A ) )
4 frsuc 6449 . . . . . . . 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 5542 . . . . . . . . . 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 15 . . . . . . . . 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 5526 . . . . . . . . 9  |-  ( Q `
 suc  A )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  A
)
96, 8syl6eqr 2333 . . . . . . . 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 5542 . . . . . . . . . 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 5526 . . . . . . . . . 10  |-  ( Q `
 A )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  A )
1210, 11syl6eqr 2333 . . . . . . . . 9  |-  ( A  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  A )  =  ( Q `  A ) )
1312fveq2d 5529 . . . . . . . 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 2323 . . . . . . 7  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  A
) ) )
157seqomlem1 6462 . . . . . . . 8  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
1615fveq2d 5529 . . . . . . 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 5861 . . . . . . . 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 5539 . . . . . . . . . 10  |-  ( 2nd `  ( Q `  A
) )  e.  _V
19 suceq 4457 . . . . . . . . . . . 12  |-  ( i  =  A  ->  suc  i  =  suc  A )
20 oveq1 5865 . . . . . . . . . . . 12  |-  ( i  =  A  ->  (
i F v )  =  ( A F v ) )
2119, 20opeq12d 3804 . . . . . . . . . . 11  |-  ( i  =  A  ->  <. suc  i ,  ( i F v ) >.  =  <. suc 
A ,  ( A F v ) >.
)
22 oveq2 5866 . . . . . . . . . . . 12  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  ( A F v )  =  ( A F ( 2nd `  ( Q `
 A ) ) ) )
2322opeq2d 3803 . . . . . . . . . . 11  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  <. suc  A ,  ( A F v ) >.  =  <. suc 
A ,  ( A F ( 2nd `  ( Q `  A )
) ) >. )
24 eqid 2283 . . . . . . . . . . 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 4237 . . . . . . . . . . 11  |-  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  e.  _V
2621, 23, 24, 25ovmpt2 5983 . . . . . . . . . 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 652 . . . . . . . . 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 5542 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  ( Q `  A ) )
2928, 15eqtrd 2315 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  <. A ,  ( 2nd `  ( Q `  A )
) >. )
30 frfnom 6447 . . . . . . . . . . . . . . . . . 18  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )  |` 
om )  Fn  om
317reseq1i 4951 . . . . . . . . . . . . . . . . . . 19  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
3231fneq1i 5338 . . . . . . . . . . . . . . . . . 18  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
3330, 32mpbir 200 . . . . . . . . . . . . . . . . 17  |-  ( Q  |`  om )  Fn  om
34 fnfvelrn 5662 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Q  |`  om )  Fn  om  /\  A  e. 
om )  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3533, 34mpan 651 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3629, 35eqeltrrd 2358 . . . . . . . . . . . . . . 15  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ran  ( Q  |`  om ) )
37 df-ima 4702 . . . . . . . . . . . . . . 15  |-  ( Q
" om )  =  ran  ( Q  |`  om )
3836, 37syl6eleqr 2374 . . . . . . . . . . . . . 14  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ( Q
" om ) )
39 df-br 4024 . . . . . . . . . . . . . 14  |-  ( A ( Q " om ) ( 2nd `  ( Q `  A )
)  <->  <. A ,  ( 2nd `  ( Q `
 A ) )
>.  e.  ( Q " om ) )
4038, 39sylibr 203 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  A
( Q " om ) ( 2nd `  ( Q `  A )
) )
417seqomlem2 6463 . . . . . . . . . . . . . 14  |-  ( Q
" om )  Fn 
om
42 fnbrfvb 5563 . . . . . . . . . . . . . 14  |-  ( ( ( Q " om )  Fn  om  /\  A  e.  om )  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4341, 42mpan 651 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4440, 43mpbird 223 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  (
( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
) )
4544eqcomd 2288 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( 2nd `  ( Q `  A ) )  =  ( ( Q " om ) `  A ) )
4645oveq2d 5874 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( A F ( 2nd `  ( Q `  A )
) )  =  ( A F ( ( Q " om ) `  A ) ) )
4746opeq2d 3803 . . . . . . . . 9  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  =  <. suc 
A ,  ( A F ( ( Q
" om ) `  A ) ) >.
)
4827, 47eqtrd 2315 . . . . . . . 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 2329 . . . . . . 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 2319 . . . . . 6  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >. )
513, 50eqtrd 2315 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  = 
<. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.
)
52 fnfvelrn 5662 . . . . . 6  |-  ( ( ( Q  |`  om )  Fn  om  /\  suc  A  e.  om )  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5333, 1, 52sylancr 644 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5451, 53eqeltrrd 2358 . . . 4  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ran  ( Q  |`  om )
)
5554, 37syl6eleqr 2374 . . 3  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ( Q " om ) )
56 df-br 4024 . . 3  |-  ( suc 
A ( Q " om ) ( A F ( ( Q " om ) `  A ) )  <->  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >.  e.  ( Q
" om ) )
5755, 56sylibr 203 . 2  |-  ( A  e.  om  ->  suc  A ( Q " om ) ( A F ( ( Q " om ) `  A ) ) )
58 fnbrfvb 5563 . . 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 644 . 2  |-  ( A  e.  om  ->  (
( ( Q " om ) `  suc  A
)  =  ( A F ( ( Q
" om ) `  A ) )  <->  suc  A ( Q " om )
( A F ( ( Q " om ) `  A )
) ) )
6057, 59mpbird 223 1  |-  ( A  e.  om  ->  (
( Q " om ) `  suc  A )  =  ( A F ( ( Q " om ) `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643   class class class wbr 4023    _I cid 4304   suc csuc 4394   omcom 4656   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121   reccrdg 6422
This theorem is referenced by:  seqomsuc  6469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-recs 6388  df-rdg 6423
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