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Theorem seqomlem2 7123
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 7107 . . . . . . 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 5063 . . . . . . . 8  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
43fneq1i 5631 . . . . . . 7  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
51, 4mpbir 212 . . . . . 6  |-  ( Q  |`  om )  Fn  om
6 fvres 5839 . . . . . . . . 9  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  ( Q `  b ) )
72seqomlem1 7122 . . . . . . . . 9  |-  ( b  e.  om  ->  ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>. )
86, 7eqtrd 2462 . . . . . . . 8  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
9 fvex 5835 . . . . . . . . 9  |-  ( 2nd `  ( Q `  b
) )  e.  _V
10 opelxpi 4828 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( 2nd `  ( Q `
 b ) )  e.  _V )  ->  <. b ,  ( 2nd `  ( Q `  b
) ) >.  e.  ( om  X.  _V )
)
119, 10mpan2 675 . . . . . . . 8  |-  ( b  e.  om  ->  <. b ,  ( 2nd `  ( Q `  b )
) >.  e.  ( om 
X.  _V ) )
128, 11eqeltrd 2506 . . . . . . 7  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  e.  ( om  X.  _V )
)
1312rgen 2724 . . . . . 6  |-  A. b  e.  om  ( ( Q  |`  om ) `  b
)  e.  ( om 
X.  _V )
14 ffnfv 6008 . . . . . 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 928 . . . . 5  |-  ( Q  |`  om ) : om --> ( om  X.  _V )
16 frn 5695 . . . . 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 4367 . . . . . . . . . 10  |-  ( a ran  ( Q  |`  om ) b  <->  <. a ,  b >.  e.  ran  ( Q  |`  om )
)
19 fvelrnb 5872 . . . . . . . . . . 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 5839 . . . . . . . . . . . 12  |-  ( c  e.  om  ->  (
( Q  |`  om ) `  c )  =  ( Q `  c ) )
2221eqeq1d 2430 . . . . . . . . . . 11  |-  ( c  e.  om  ->  (
( ( Q  |`  om ) `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  b >. )
)
2322rexbiia 2865 . . . . . . . . . 10  |-  ( E. c  e.  om  (
( Q  |`  om ) `  c )  =  <. a ,  b >.  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
2418, 20, 233bitri 274 . . . . . . . . 9  |-  ( a ran  ( Q  |`  om ) b  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
252seqomlem1 7122 . . . . . . . . . . . . . . . 16  |-  ( c  e.  om  ->  ( Q `  c )  =  <. c ,  ( 2nd `  ( Q `
 c ) )
>. )
2625adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( Q `  c
)  =  <. c ,  ( 2nd `  ( Q `  c )
) >. )
2726eqeq1d 2430 . . . . . . . . . . . . . 14  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  <->  <. c ,  ( 2nd `  ( Q `  c )
) >.  =  <. a ,  b >. )
)
28 vex 3025 . . . . . . . . . . . . . . 15  |-  c  e. 
_V
29 fvex 5835 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( Q `  c
) )  e.  _V
3028, 29opth1 4637 . . . . . . . . . . . . . 14  |-  ( <.
c ,  ( 2nd `  ( Q `  c
) ) >.  =  <. a ,  b >.  ->  c  =  a )
3127, 30syl6bi 231 . . . . . . . . . . . . 13  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  c  =  a ) )
32 fveq2 5825 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  ( Q `  c )  =  ( Q `  a ) )
3332eqeq1d 2430 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  <->  ( Q `  a )  =  <. a ,  b >. )
)
3433biimpd 210 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
3531, 34syli 38 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
36 fveq2 5825 . . . . . . . . . . . . 13  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  ( 2nd `  ( Q `  a )
)  =  ( 2nd `  <. a ,  b
>. ) )
37 vex 3025 . . . . . . . . . . . . . 14  |-  a  e. 
_V
38 vex 3025 . . . . . . . . . . . . . 14  |-  b  e. 
_V
3937, 38op2nd 6760 . . . . . . . . . . . . 13  |-  ( 2nd `  <. a ,  b
>. )  =  b
4036, 39syl6req 2479 . . . . . . . . . . . 12  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) )
4135, 40syl6 34 . . . . . . . . . . 11  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  b  =  ( 2nd `  ( Q `  a )
) ) )
4241rexlimdva 2856 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
432seqomlem1 7122 . . . . . . . . . . . 12  |-  ( a  e.  om  ->  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )
4432eqeq1d 2430 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4544rspcev 3125 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4643, 45mpdan 672 . . . . . . . . . . 11  |-  ( a  e.  om  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
47 opeq2 4131 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  <. a ,  b >.  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4847eqeq2d 2438 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  ( ( Q `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4948rexbidv 2878 . . . . . . . . . . 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 225 . . . . . . . . . 10  |-  ( a  e.  om  ->  (
b  =  ( 2nd `  ( Q `  a
) )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
)
5142, 50impbid 193 . . . . . . . . 9  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>. 
<->  b  =  ( 2nd `  ( Q `  a
) ) ) )
5224, 51syl5bb 260 . . . . . . . 8  |-  ( a  e.  om  ->  (
a ran  ( Q  |` 
om ) b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5352alrimiv 1767 . . . . . . 7  |-  ( a  e.  om  ->  A. b
( a ran  ( Q  |`  om ) b  <-> 
b  =  ( 2nd `  ( Q `  a
) ) ) )
54 fvex 5835 . . . . . . . 8  |-  ( 2nd `  ( Q `  a
) )  e.  _V
55 eqeq2 2439 . . . . . . . . . 10  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( b  =  c  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5655bibi2d 319 . . . . . . . . 9  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( (
a ran  ( Q  |` 
om ) b  <->  b  =  c )  <->  ( a ran  ( Q  |`  om )
b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) ) )
5756albidv 1761 . . . . . . . 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 3116 . . . . . . 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 2280 . . . . . 6  |-  ( E! b  a ran  ( Q  |`  om ) b  <->  E. c A. b ( a ran  ( Q  |`  om ) b  <->  b  =  c ) )
6159, 60sylibr 215 . . . . 5  |-  ( a  e.  om  ->  E! b  a ran  ( Q  |`  om ) b )
6261rgen 2724 . . . 4  |-  A. a  e.  om  E! b  a ran  ( Q  |`  om ) b
63 dff3 5994 . . . 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 928 . . 3  |-  ran  ( Q  |`  om ) : om --> _V
65 df-ima 4809 . . . 4  |-  ( Q
" om )  =  ran  ( Q  |`  om )
6665feq1i 5681 . . 3  |-  ( ( Q " om ) : om --> _V  <->  ran  ( Q  |`  om ) : om --> _V )
6764, 66mpbir 212 . 2  |-  ( Q
" om ) : om --> _V
68 dffn2 5690 . 2  |-  ( ( Q " om )  Fn  om  <->  ( Q " om ) : om --> _V )
6967, 68mpbir 212 1  |-  ( Q
" om )  Fn 
om
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   E!weu 2276   A.wral 2714   E.wrex 2715   _Vcvv 3022    C_ wss 3379   (/)c0 3704   <.cop 3947   class class class wbr 4366    _I cid 4706    X. cxp 4794   ran crn 4797    |` cres 4798   "cima 4799   suc csuc 5387    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   omcom 6650   2ndc2nd 6750   reccrdg 7082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083
This theorem is referenced by:  seqomlem3  7124  seqomlem4  7125  fnseqom  7127
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