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Theorem seqomlem4 6481
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 4692 . . . . . . 7  |-  ( A  e.  om  ->  suc  A  e.  om )
2 fvres 5558 . . . . . . 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 6465 . . . . . . . 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 5558 . . . . . . . . . 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 5542 . . . . . . . . 9  |-  ( Q `
 suc  A )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  A
)
96, 8syl6eqr 2346 . . . . . . . 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 5558 . . . . . . . . . 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 5542 . . . . . . . . . 10  |-  ( Q `
 A )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  A )
1210, 11syl6eqr 2346 . . . . . . . . 9  |-  ( A  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  A )  =  ( Q `  A ) )
1312fveq2d 5545 . . . . . . . 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 2336 . . . . . . 7  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  A
) ) )
157seqomlem1 6478 . . . . . . . 8  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
1615fveq2d 5545 . . . . . . 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 5877 . . . . . . . 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 5555 . . . . . . . . . 10  |-  ( 2nd `  ( Q `  A
) )  e.  _V
19 suceq 4473 . . . . . . . . . . . 12  |-  ( i  =  A  ->  suc  i  =  suc  A )
20 oveq1 5881 . . . . . . . . . . . 12  |-  ( i  =  A  ->  (
i F v )  =  ( A F v ) )
2119, 20opeq12d 3820 . . . . . . . . . . 11  |-  ( i  =  A  ->  <. suc  i ,  ( i F v ) >.  =  <. suc 
A ,  ( A F v ) >.
)
22 oveq2 5882 . . . . . . . . . . . 12  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  ( A F v )  =  ( A F ( 2nd `  ( Q `
 A ) ) ) )
2322opeq2d 3819 . . . . . . . . . . 11  |-  ( v  =  ( 2nd `  ( Q `  A )
)  ->  <. suc  A ,  ( A F v ) >.  =  <. suc 
A ,  ( A F ( 2nd `  ( Q `  A )
) ) >. )
24 eqid 2296 . . . . . . . . . . 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 4253 . . . . . . . . . . 11  |-  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  e.  _V
2621, 23, 24, 25ovmpt2 5999 . . . . . . . . . 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 5558 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  ( Q `  A ) )
2928, 15eqtrd 2328 . . . . . . . . . . . . . . . 16  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  A )  =  <. A ,  ( 2nd `  ( Q `  A )
) >. )
30 frfnom 6463 . . . . . . . . . . . . . . . . . 18  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )  |` 
om )  Fn  om
317reseq1i 4967 . . . . . . . . . . . . . . . . . . 19  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
3231fneq1i 5354 . . . . . . . . . . . . . . . . . 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 5678 . . . . . . . . . . . . . . . . 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 2371 . . . . . . . . . . . . . . 15  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ran  ( Q  |`  om ) )
37 df-ima 4718 . . . . . . . . . . . . . . 15  |-  ( Q
" om )  =  ran  ( Q  |`  om )
3836, 37syl6eleqr 2387 . . . . . . . . . . . . . 14  |-  ( A  e.  om  ->  <. A , 
( 2nd `  ( Q `  A )
) >.  e.  ( Q
" om ) )
39 df-br 4040 . . . . . . . . . . . . . 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 6479 . . . . . . . . . . . . . 14  |-  ( Q
" om )  Fn 
om
42 fnbrfvb 5579 . . . . . . . . . . . . . 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 2301 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( 2nd `  ( Q `  A ) )  =  ( ( Q " om ) `  A ) )
4645oveq2d 5890 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( A F ( 2nd `  ( Q `  A )
) )  =  ( A F ( ( Q " om ) `  A ) ) )
4746opeq2d 3819 . . . . . . . . 9  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( 2nd `  ( Q `  A )
) ) >.  =  <. suc 
A ,  ( A F ( ( Q
" om ) `  A ) ) >.
)
4827, 47eqtrd 2328 . . . . . . . 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 2342 . . . . . . 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 2332 . . . . . 6  |-  ( A  e.  om  ->  ( Q `  suc  A )  =  <. suc  A , 
( A F ( ( Q " om ) `  A )
) >. )
513, 50eqtrd 2328 . . . . 5  |-  ( A  e.  om  ->  (
( Q  |`  om ) `  suc  A )  = 
<. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.
)
52 fnfvelrn 5678 . . . . . 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 2371 . . . 4  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ran  ( Q  |`  om )
)
5554, 37syl6eleqr 2387 . . 3  |-  ( A  e.  om  ->  <. suc  A ,  ( A F ( ( Q " om ) `  A ) ) >.  e.  ( Q " om ) )
56 df-br 4040 . . 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 5579 . . 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 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656   class class class wbr 4039    _I cid 4320   suc csuc 4410   omcom 4672   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   reccrdg 6438
This theorem is referenced by:  seqomsuc  6485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-recs 6404  df-rdg 6439
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