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Theorem seqomlem2 6637
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 6621 . . . . . . 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 5075 . . . . . . . 8  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
43fneq1i 5472 . . . . . . 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 5678 . . . . . . . . 9  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  ( Q `  b ) )
72seqomlem1 6636 . . . . . . . . 9  |-  ( b  e.  om  ->  ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>. )
86, 7eqtrd 2412 . . . . . . . 8  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
9 fvex 5675 . . . . . . . . 9  |-  ( 2nd `  ( Q `  b
) )  e.  _V
10 opelxpi 4843 . . . . . . . . 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 2454 . . . . . . 7  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  e.  ( om  X.  _V )
)
1312rgen 2707 . . . . . 6  |-  A. b  e.  om  ( ( Q  |`  om ) `  b
)  e.  ( om 
X.  _V )
14 ffnfv 5826 . . . . . 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 5530 . . . . 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 4147 . . . . . . . . . 10  |-  ( a ran  ( Q  |`  om ) b  <->  <. a ,  b >.  e.  ran  ( Q  |`  om )
)
19 fvelrnb 5706 . . . . . . . . . . 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 5678 . . . . . . . . . . . 12  |-  ( c  e.  om  ->  (
( Q  |`  om ) `  c )  =  ( Q `  c ) )
2221eqeq1d 2388 . . . . . . . . . . 11  |-  ( c  e.  om  ->  (
( ( Q  |`  om ) `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  b >. )
)
2322rexbiia 2675 . . . . . . . . . 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 6636 . . . . . . . . . . . . . . . 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 2388 . . . . . . . . . . . . . 14  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  <->  <. c ,  ( 2nd `  ( Q `  c )
) >.  =  <. a ,  b >. )
)
28 vex 2895 . . . . . . . . . . . . . . 15  |-  c  e. 
_V
29 fvex 5675 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( Q `  c
) )  e.  _V
3028, 29opth1 4368 . . . . . . . . . . . . . 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 5661 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  ( Q `  c )  =  ( Q `  a ) )
3332eqeq1d 2388 . . . . . . . . . . . . . 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 5661 . . . . . . . . . . . . 13  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  ( 2nd `  ( Q `  a )
)  =  ( 2nd `  <. a ,  b
>. ) )
37 vex 2895 . . . . . . . . . . . . . 14  |-  a  e. 
_V
38 vex 2895 . . . . . . . . . . . . . 14  |-  b  e. 
_V
3937, 38op2nd 6288 . . . . . . . . . . . . 13  |-  ( 2nd `  <. a ,  b
>. )  =  b
4036, 39syl6req 2429 . . . . . . . . . . . 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 2766 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
432seqomlem1 6636 . . . . . . . . . . . 12  |-  ( a  e.  om  ->  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )
4432eqeq1d 2388 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4544rspcev 2988 . . . . . . . . . . . 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 3920 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  <. a ,  b >.  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4847eqeq2d 2391 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  ( ( Q `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4948rexbidv 2663 . . . . . . . . . . 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 1638 . . . . . . 7  |-  ( a  e.  om  ->  A. b
( a ran  ( Q  |`  om ) b  <-> 
b  =  ( 2nd `  ( Q `  a
) ) ) )
54 fvex 5675 . . . . . . . 8  |-  ( 2nd `  ( Q `  a
) )  e.  _V
55 eqeq2 2389 . . . . . . . . . 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 1632 . . . . . . . 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 2979 . . . . . . 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 2235 . . . . . 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 2707 . . . 4  |-  A. a  e.  om  E! b  a ran  ( Q  |`  om ) b
63 dff3 5814 . . . 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 4824 . . . 4  |-  ( Q
" om )  =  ran  ( Q  |`  om )
6665feq1i 5518 . . 3  |-  ( ( Q " om ) : om --> _V  <->  ran  ( Q  |`  om ) : om --> _V )
6764, 66mpbir 201 . 2  |-  ( Q
" om ) : om --> _V
68 dffn2 5525 . 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 1546   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2231   A.wral 2642   E.wrex 2643   _Vcvv 2892    C_ wss 3256   (/)c0 3564   <.cop 3753   class class class wbr 4146    _I cid 4427   suc csuc 4517   omcom 4778    X. cxp 4809   ran crn 4812    |` cres 4813   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   2ndc2nd 6280   reccrdg 6596
This theorem is referenced by:  seqomlem3  6638  seqomlem4  6639  fnseqom  6641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-recs 6562  df-rdg 6597
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