MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqomlem2 Structured version   Visualization version   Unicode version

Theorem seqomlem2 7186
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 7170 . . . . . . 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 5107 . . . . . . . 8  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
43fneq1i 5680 . . . . . . 7  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
51, 4mpbir 214 . . . . . 6  |-  ( Q  |`  om )  Fn  om
6 fvres 5893 . . . . . . . . 9  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  ( Q `  b ) )
72seqomlem1 7185 . . . . . . . . 9  |-  ( b  e.  om  ->  ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>. )
86, 7eqtrd 2505 . . . . . . . 8  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
9 fvex 5889 . . . . . . . . 9  |-  ( 2nd `  ( Q `  b
) )  e.  _V
10 opelxpi 4871 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( 2nd `  ( Q `
 b ) )  e.  _V )  ->  <. b ,  ( 2nd `  ( Q `  b
) ) >.  e.  ( om  X.  _V )
)
119, 10mpan2 685 . . . . . . . 8  |-  ( b  e.  om  ->  <. b ,  ( 2nd `  ( Q `  b )
) >.  e.  ( om 
X.  _V ) )
128, 11eqeltrd 2549 . . . . . . 7  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  e.  ( om  X.  _V )
)
1312rgen 2766 . . . . . 6  |-  A. b  e.  om  ( ( Q  |`  om ) `  b
)  e.  ( om 
X.  _V )
14 ffnfv 6064 . . . . . 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 934 . . . . 5  |-  ( Q  |`  om ) : om --> ( om  X.  _V )
16 frn 5747 . . . . 5  |-  ( ( Q  |`  om ) : om --> ( om  X.  _V )  ->  ran  ( Q  |`  om )  C_  ( om  X.  _V )
)
1715, 16ax-mp 5 . . . 4  |-  ran  ( Q  |`  om )  C_  ( om  X.  _V )
18 df-br 4396 . . . . . . . . . 10  |-  ( a ran  ( Q  |`  om ) b  <->  <. a ,  b >.  e.  ran  ( Q  |`  om )
)
19 fvelrnb 5926 . . . . . . . . . . 11  |-  ( ( Q  |`  om )  Fn  om  ->  ( <. a ,  b >.  e.  ran  ( Q  |`  om )  <->  E. c  e.  om  (
( Q  |`  om ) `  c )  =  <. a ,  b >. )
)
205, 19ax-mp 5 . . . . . . . . . 10  |-  ( <.
a ,  b >.  e.  ran  ( Q  |`  om )  <->  E. c  e.  om  ( ( Q  |`  om ) `  c )  =  <. a ,  b
>. )
21 fvres 5893 . . . . . . . . . . . 12  |-  ( c  e.  om  ->  (
( Q  |`  om ) `  c )  =  ( Q `  c ) )
2221eqeq1d 2473 . . . . . . . . . . 11  |-  ( c  e.  om  ->  (
( ( Q  |`  om ) `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  b >. )
)
2322rexbiia 2880 . . . . . . . . . 10  |-  ( E. c  e.  om  (
( Q  |`  om ) `  c )  =  <. a ,  b >.  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
2418, 20, 233bitri 279 . . . . . . . . 9  |-  ( a ran  ( Q  |`  om ) b  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
252seqomlem1 7185 . . . . . . . . . . . . . . . 16  |-  ( c  e.  om  ->  ( Q `  c )  =  <. c ,  ( 2nd `  ( Q `
 c ) )
>. )
2625adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( Q `  c
)  =  <. c ,  ( 2nd `  ( Q `  c )
) >. )
2726eqeq1d 2473 . . . . . . . . . . . . . 14  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  <->  <. c ,  ( 2nd `  ( Q `  c )
) >.  =  <. a ,  b >. )
)
28 vex 3034 . . . . . . . . . . . . . . 15  |-  c  e. 
_V
29 fvex 5889 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( Q `  c
) )  e.  _V
3028, 29opth1 4675 . . . . . . . . . . . . . 14  |-  ( <.
c ,  ( 2nd `  ( Q `  c
) ) >.  =  <. a ,  b >.  ->  c  =  a )
3127, 30syl6bi 236 . . . . . . . . . . . . 13  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  c  =  a ) )
32 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  ( Q `  c )  =  ( Q `  a ) )
3332eqeq1d 2473 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  <->  ( Q `  a )  =  <. a ,  b >. )
)
3433biimpd 212 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
3531, 34syli 37 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
36 fveq2 5879 . . . . . . . . . . . . 13  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  ( 2nd `  ( Q `  a )
)  =  ( 2nd `  <. a ,  b
>. ) )
37 vex 3034 . . . . . . . . . . . . . 14  |-  a  e. 
_V
38 vex 3034 . . . . . . . . . . . . . 14  |-  b  e. 
_V
3937, 38op2nd 6821 . . . . . . . . . . . . 13  |-  ( 2nd `  <. a ,  b
>. )  =  b
4036, 39syl6req 2522 . . . . . . . . . . . 12  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) )
4135, 40syl6 33 . . . . . . . . . . 11  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  b  =  ( 2nd `  ( Q `  a )
) ) )
4241rexlimdva 2871 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
432seqomlem1 7185 . . . . . . . . . . . 12  |-  ( a  e.  om  ->  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )
4432eqeq1d 2473 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4544rspcev 3136 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4643, 45mpdan 681 . . . . . . . . . . 11  |-  ( a  e.  om  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
47 opeq2 4159 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  <. a ,  b >.  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4847eqeq2d 2481 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  ( ( Q `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4948rexbidv 2892 . . . . . . . . . . 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 230 . . . . . . . . . 10  |-  ( a  e.  om  ->  (
b  =  ( 2nd `  ( Q `  a
) )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
)
5142, 50impbid 195 . . . . . . . . 9  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>. 
<->  b  =  ( 2nd `  ( Q `  a
) ) ) )
5224, 51syl5bb 265 . . . . . . . 8  |-  ( a  e.  om  ->  (
a ran  ( Q  |` 
om ) b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5352alrimiv 1781 . . . . . . 7  |-  ( a  e.  om  ->  A. b
( a ran  ( Q  |`  om ) b  <-> 
b  =  ( 2nd `  ( Q `  a
) ) ) )
54 fvex 5889 . . . . . . . 8  |-  ( 2nd `  ( Q `  a
) )  e.  _V
55 eqeq2 2482 . . . . . . . . . 10  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( b  =  c  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5655bibi2d 325 . . . . . . . . 9  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( (
a ran  ( Q  |` 
om ) b  <->  b  =  c )  <->  ( a ran  ( Q  |`  om )
b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) ) )
5756albidv 1775 . . . . . . . 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 3127 . . . . . . 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 17 . . . . . 6  |-  ( a  e.  om  ->  E. c A. b ( a ran  ( Q  |`  om )
b  <->  b  =  c ) )
60 df-eu 2323 . . . . . 6  |-  ( E! b  a ran  ( Q  |`  om ) b  <->  E. c A. b ( a ran  ( Q  |`  om ) b  <->  b  =  c ) )
6159, 60sylibr 217 . . . . 5  |-  ( a  e.  om  ->  E! b  a ran  ( Q  |`  om ) b )
6261rgen 2766 . . . 4  |-  A. a  e.  om  E! b  a ran  ( Q  |`  om ) b
63 dff3 6050 . . . 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 934 . . 3  |-  ran  ( Q  |`  om ) : om --> _V
65 df-ima 4852 . . . 4  |-  ( Q
" om )  =  ran  ( Q  |`  om )
6665feq1i 5730 . . 3  |-  ( ( Q " om ) : om --> _V  <->  ran  ( Q  |`  om ) : om --> _V )
6764, 66mpbir 214 . 2  |-  ( Q
" om ) : om --> _V
68 dffn2 5741 . 2  |-  ( ( Q " om )  Fn  om  <->  ( Q " om ) : om --> _V )
6967, 68mpbir 214 1  |-  ( Q
" om )  Fn 
om
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   A.wral 2756   E.wrex 2757   _Vcvv 3031    C_ wss 3390   (/)c0 3722   <.cop 3965   class class class wbr 4395    _I cid 4749    X. cxp 4837   ran crn 4840    |` cres 4841   "cima 4842   suc csuc 5432    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   omcom 6711   2ndc2nd 6811   reccrdg 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146
This theorem is referenced by:  seqomlem3  7187  seqomlem4  7188  fnseqom  7190
  Copyright terms: Public domain W3C validator