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Theorem seqomlem4 6712
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 4867 . . . . . . 7  |-  ( A  e.  om  ->  suc  A  e.  om )
2 fvres 5747 . . . . . . 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 6696 . . . . . . . 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 5747 . . . . . . . . . 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 5731 . . . . . . . . 9  |-  ( Q `
 suc  A )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  A
)
96, 8syl6eqr 2488 . . . . . . . 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 5747 . . . . . . . . . 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 5731 . . . . . . . . . 10  |-  ( Q `
 A )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  A )
1210, 11syl6eqr 2488 . . . . . . . . 9  |-  ( A  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  A )  =  ( Q `  A ) )
1312fveq2d 5734 . . . . . . . 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 2478 . . . . . . 7  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  A
) ) )
157seqomlem1 6709 . . . . . . . 8  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
1615fveq2d 5734 . . . . . . 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 6086 . . . . . . . 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 5744 . . . . . . . . . 10  |-  ( 2nd `  ( Q `  A
) )  e.  _V
19 suceq 4648 . . . . . . . . . . . 12  |-  ( i  =  A  ->  suc  i  =  suc  A )
20 oveq1 6090 . . . . . . . . . . . 12  |-  ( i  =  A  ->  (
i F v )  =  ( A F v ) )
2119, 20opeq12d 3994 . . . . . . . . . . 11  |-  ( i  =  A  ->  <. suc  i ,  ( i F v ) >.  =  <. suc 
A ,  ( A F v ) >.
)
22 oveq2 6091 . . . . . . . . . . . 12  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  ( A F v )  =  ( A F ( 2nd `  ( Q `
 A ) ) ) )
2322opeq2d 3993 . . . . . . . . . . 11  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  <. suc  A ,  ( A F v ) >.  =  <. suc 
A ,  ( A F ( 2nd `  ( Q `  A )
) ) >. )
24 eqid 2438 . . . . . . . . . . 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 4429 . . . . . . . . . . 11  |-  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  e.  _V
2621, 23, 24, 25ovmpt2 6211 . . . . . . . . . 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 654 . . . . . . . . 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 5747 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  ( Q `  A ) )
2928, 15eqtrd 2470 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  <. A ,  ( 2nd `  ( Q `  A )
) >. )
30 frfnom 6694 . . . . . . . . . . . . . . . . . 18  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )  |` 
om )  Fn  om
317reseq1i 5144 . . . . . . . . . . . . . . . . . . 19  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
3231fneq1i 5541 . . . . . . . . . . . . . . . . . 18  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
3330, 32mpbir 202 . . . . . . . . . . . . . . . . 17  |-  ( Q  |`  om )  Fn  om
34 fnfvelrn 5869 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Q  |`  om )  Fn  om  /\  A  e. 
om )  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3533, 34mpan 653 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  e.  ran  ( Q  |`  om )
)
3629, 35eqeltrrd 2513 . . . . . . . . . . . . . . 15  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ran  ( Q  |`  om ) )
37 df-ima 4893 . . . . . . . . . . . . . . 15  |-  ( Q
" om )  =  ran  ( Q  |`  om )
3836, 37syl6eleqr 2529 . . . . . . . . . . . . . 14  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ( Q
" om ) )
39 df-br 4215 . . . . . . . . . . . . . 14  |-  ( A ( Q " om ) ( 2nd `  ( Q `  A )
)  <->  <. A ,  ( 2nd `  ( Q `
 A ) )
>.  e.  ( Q " om ) )
4038, 39sylibr 205 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  A
( Q " om ) ( 2nd `  ( Q `  A )
) )
417seqomlem2 6710 . . . . . . . . . . . . . 14  |-  ( Q
" om )  Fn 
om
42 fnbrfvb 5769 . . . . . . . . . . . . . 14  |-  ( ( ( Q " om )  Fn  om  /\  A  e.  om )  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4341, 42mpan 653 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  (
( ( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
)  <->  A ( Q " om ) ( 2nd `  ( Q `  A )
) ) )
4440, 43mpbird 225 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  (
( Q " om ) `  A )  =  ( 2nd `  ( Q `  A )
) )
4544eqcomd 2443 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( 2nd `  ( Q `  A ) )  =  ( ( Q " om ) `  A ) )
4645oveq2d 6099 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( A F ( 2nd `  ( Q `  A )
) )  =  ( A F ( ( Q " om ) `  A ) ) )
4746opeq2d 3993 . . . . . . . . 9  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  =  <. suc 
A ,  ( A F ( ( Q
" om ) `  A ) ) >.
)
4827, 47eqtrd 2470 . . . . . . . 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 2484 . . . . . . 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 2474 . . . . . 6  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >. )
513, 50eqtrd 2470 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  = 
<. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.
)
52 fnfvelrn 5869 . . . . . 6  |-  ( ( ( Q  |`  om )  Fn  om  /\  suc  A  e.  om )  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5333, 1, 52sylancr 646 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  e. 
ran  ( Q  |`  om ) )
5451, 53eqeltrrd 2513 . . . 4  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ran  ( Q  |`  om )
)
5554, 37syl6eleqr 2529 . . 3  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ( Q " om ) )
56 df-br 4215 . . 3  |-  ( suc 
A ( Q " om ) ( A F ( ( Q " om ) `  A ) )  <->  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >.  e.  ( Q
" om ) )
5755, 56sylibr 205 . 2  |-  ( A  e.  om  ->  suc  A ( Q " om ) ( A F ( ( Q " om ) `  A ) ) )
58 fnbrfvb 5769 . . 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 646 . 2  |-  ( A  e.  om  ->  (
( ( Q " om ) `  suc  A
)  =  ( A F ( ( Q
" om ) `  A ) )  <->  suc  A ( Q " om )
( A F ( ( Q " om ) `  A )
) ) )
6057, 59mpbird 225 1  |-  ( A  e.  om  ->  (
( Q " om ) `  suc  A )  =  ( A F ( ( Q " om ) `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   <.cop 3819   class class class wbr 4214    _I cid 4495   suc csuc 4585   omcom 4847   ran crn 4881    |` cres 4882   "cima 4883    Fn wfn 5451   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   2ndc2nd 6350   reccrdg 6669
This theorem is referenced by:  seqomsuc  6716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-recs 6635  df-rdg 6670
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