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Theorem seqomlem2 6700
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
seqomlem2  |-  ( Q
" om )  Fn 
om
Distinct variable groups:    Q, i,
v    i, F, v
Allowed substitution hints:    I( v, i)

Proof of Theorem seqomlem2
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frfnom 6684 . . . . . . 7  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )  |` 
om )  Fn  om
2 seqomlem.a . . . . . . . . 9  |-  Q  =  rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)
32reseq1i 5134 . . . . . . . 8  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
43fneq1i 5531 . . . . . . 7  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
51, 4mpbir 201 . . . . . 6  |-  ( Q  |`  om )  Fn  om
6 fvres 5737 . . . . . . . . 9  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  ( Q `  b ) )
72seqomlem1 6699 . . . . . . . . 9  |-  ( b  e.  om  ->  ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>. )
86, 7eqtrd 2467 . . . . . . . 8  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
9 fvex 5734 . . . . . . . . 9  |-  ( 2nd `  ( Q `  b
) )  e.  _V
10 opelxpi 4902 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( 2nd `  ( Q `
 b ) )  e.  _V )  ->  <. b ,  ( 2nd `  ( Q `  b
) ) >.  e.  ( om  X.  _V )
)
119, 10mpan2 653 . . . . . . . 8  |-  ( b  e.  om  ->  <. b ,  ( 2nd `  ( Q `  b )
) >.  e.  ( om 
X.  _V ) )
128, 11eqeltrd 2509 . . . . . . 7  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  e.  ( om  X.  _V )
)
1312rgen 2763 . . . . . 6  |-  A. b  e.  om  ( ( Q  |`  om ) `  b
)  e.  ( om 
X.  _V )
14 ffnfv 5886 . . . . . 6  |-  ( ( Q  |`  om ) : om --> ( om  X.  _V )  <->  ( ( Q  |`  om )  Fn  om  /\ 
A. b  e.  om  ( ( Q  |`  om ) `  b )  e.  ( om  X.  _V ) ) )
155, 13, 14mpbir2an 887 . . . . 5  |-  ( Q  |`  om ) : om --> ( om  X.  _V )
16 frn 5589 . . . . 5  |-  ( ( Q  |`  om ) : om --> ( om  X.  _V )  ->  ran  ( Q  |`  om )  C_  ( om  X.  _V )
)
1715, 16ax-mp 8 . . . 4  |-  ran  ( Q  |`  om )  C_  ( om  X.  _V )
18 df-br 4205 . . . . . . . . . 10  |-  ( a ran  ( Q  |`  om ) b  <->  <. a ,  b >.  e.  ran  ( Q  |`  om )
)
19 fvelrnb 5766 . . . . . . . . . . 11  |-  ( ( Q  |`  om )  Fn  om  ->  ( <. a ,  b >.  e.  ran  ( Q  |`  om )  <->  E. c  e.  om  (
( Q  |`  om ) `  c )  =  <. a ,  b >. )
)
205, 19ax-mp 8 . . . . . . . . . 10  |-  ( <.
a ,  b >.  e.  ran  ( Q  |`  om )  <->  E. c  e.  om  ( ( Q  |`  om ) `  c )  =  <. a ,  b
>. )
21 fvres 5737 . . . . . . . . . . . 12  |-  ( c  e.  om  ->  (
( Q  |`  om ) `  c )  =  ( Q `  c ) )
2221eqeq1d 2443 . . . . . . . . . . 11  |-  ( c  e.  om  ->  (
( ( Q  |`  om ) `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  b >. )
)
2322rexbiia 2730 . . . . . . . . . 10  |-  ( E. c  e.  om  (
( Q  |`  om ) `  c )  =  <. a ,  b >.  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
2418, 20, 233bitri 263 . . . . . . . . 9  |-  ( a ran  ( Q  |`  om ) b  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
252seqomlem1 6699 . . . . . . . . . . . . . . . 16  |-  ( c  e.  om  ->  ( Q `  c )  =  <. c ,  ( 2nd `  ( Q `
 c ) )
>. )
2625adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( Q `  c
)  =  <. c ,  ( 2nd `  ( Q `  c )
) >. )
2726eqeq1d 2443 . . . . . . . . . . . . . 14  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  <->  <. c ,  ( 2nd `  ( Q `  c )
) >.  =  <. a ,  b >. )
)
28 vex 2951 . . . . . . . . . . . . . . 15  |-  c  e. 
_V
29 fvex 5734 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( Q `  c
) )  e.  _V
3028, 29opth1 4426 . . . . . . . . . . . . . 14  |-  ( <.
c ,  ( 2nd `  ( Q `  c
) ) >.  =  <. a ,  b >.  ->  c  =  a )
3127, 30syl6bi 220 . . . . . . . . . . . . 13  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  c  =  a ) )
32 fveq2 5720 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  ( Q `  c )  =  ( Q `  a ) )
3332eqeq1d 2443 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  <->  ( Q `  a )  =  <. a ,  b >. )
)
3433biimpd 199 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
3531, 34syli 35 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
36 fveq2 5720 . . . . . . . . . . . . 13  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  ( 2nd `  ( Q `  a )
)  =  ( 2nd `  <. a ,  b
>. ) )
37 vex 2951 . . . . . . . . . . . . . 14  |-  a  e. 
_V
38 vex 2951 . . . . . . . . . . . . . 14  |-  b  e. 
_V
3937, 38op2nd 6348 . . . . . . . . . . . . 13  |-  ( 2nd `  <. a ,  b
>. )  =  b
4036, 39syl6req 2484 . . . . . . . . . . . 12  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) )
4135, 40syl6 31 . . . . . . . . . . 11  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  b  =  ( 2nd `  ( Q `  a )
) ) )
4241rexlimdva 2822 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
432seqomlem1 6699 . . . . . . . . . . . 12  |-  ( a  e.  om  ->  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )
4432eqeq1d 2443 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4544rspcev 3044 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4643, 45mpdan 650 . . . . . . . . . . 11  |-  ( a  e.  om  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
47 opeq2 3977 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  <. a ,  b >.  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4847eqeq2d 2446 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  ( ( Q `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4948rexbidv 2718 . . . . . . . . . . 11  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>. 
<->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. ) )
5046, 49syl5ibrcom 214 . . . . . . . . . 10  |-  ( a  e.  om  ->  (
b  =  ( 2nd `  ( Q `  a
) )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
)
5142, 50impbid 184 . . . . . . . . 9  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>. 
<->  b  =  ( 2nd `  ( Q `  a
) ) ) )
5224, 51syl5bb 249 . . . . . . . 8  |-  ( a  e.  om  ->  (
a ran  ( Q  |` 
om ) b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5352alrimiv 1641 . . . . . . 7  |-  ( a  e.  om  ->  A. b
( a ran  ( Q  |`  om ) b  <-> 
b  =  ( 2nd `  ( Q `  a
) ) ) )
54 fvex 5734 . . . . . . . 8  |-  ( 2nd `  ( Q `  a
) )  e.  _V
55 eqeq2 2444 . . . . . . . . . 10  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( b  =  c  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5655bibi2d 310 . . . . . . . . 9  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( (
a ran  ( Q  |` 
om ) b  <->  b  =  c )  <->  ( a ran  ( Q  |`  om )
b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) ) )
5756albidv 1635 . . . . . . . 8  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( A. b ( a ran  ( Q  |`  om )
b  <->  b  =  c )  <->  A. b ( a ran  ( Q  |`  om ) b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) ) )
5854, 57spcev 3035 . . . . . . 7  |-  ( A. b ( a ran  ( Q  |`  om )
b  <->  b  =  ( 2nd `  ( Q `
 a ) ) )  ->  E. c A. b ( a ran  ( Q  |`  om )
b  <->  b  =  c ) )
5953, 58syl 16 . . . . . 6  |-  ( a  e.  om  ->  E. c A. b ( a ran  ( Q  |`  om )
b  <->  b  =  c ) )
60 df-eu 2284 . . . . . 6  |-  ( E! b  a ran  ( Q  |`  om ) b  <->  E. c A. b ( a ran  ( Q  |`  om ) b  <->  b  =  c ) )
6159, 60sylibr 204 . . . . 5  |-  ( a  e.  om  ->  E! b  a ran  ( Q  |`  om ) b )
6261rgen 2763 . . . 4  |-  A. a  e.  om  E! b  a ran  ( Q  |`  om ) b
63 dff3 5874 . . . 4  |-  ( ran  ( Q  |`  om ) : om --> _V  <->  ( ran  ( Q  |`  om )  C_  ( om  X.  _V )  /\  A. a  e.  om  E! b  a ran  ( Q  |`  om )
b ) )
6417, 62, 63mpbir2an 887 . . 3  |-  ran  ( Q  |`  om ) : om --> _V
65 df-ima 4883 . . . 4  |-  ( Q
" om )  =  ran  ( Q  |`  om )
6665feq1i 5577 . . 3  |-  ( ( Q " om ) : om --> _V  <->  ran  ( Q  |`  om ) : om --> _V )
6764, 66mpbir 201 . 2  |-  ( Q
" om ) : om --> _V
68 dffn2 5584 . 2  |-  ( ( Q " om )  Fn  om  <->  ( Q " om ) : om --> _V )
6967, 68mpbir 201 1  |-  ( Q
" om )  Fn 
om
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   (/)c0 3620   <.cop 3809   class class class wbr 4204    _I cid 4485   suc csuc 4575   omcom 4837    X. cxp 4868   ran crn 4871    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   2ndc2nd 6340   reccrdg 6659
This theorem is referenced by:  seqomlem3  6701  seqomlem4  6702  fnseqom  6704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-recs 6625  df-rdg 6660
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