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Theorem seqomlem2 6479
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 6463 . . . . . . 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 4967 . . . . . . . 8  |-  ( Q  |`  om )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om )
43fneq1i 5354 . . . . . . 7  |-  ( ( Q  |`  om )  Fn  om  <->  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om )  Fn  om )
51, 4mpbir 200 . . . . . 6  |-  ( Q  |`  om )  Fn  om
6 fvres 5558 . . . . . . . . 9  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  ( Q `  b ) )
72seqomlem1 6478 . . . . . . . . 9  |-  ( b  e.  om  ->  ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>. )
86, 7eqtrd 2328 . . . . . . . 8  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
9 fvex 5555 . . . . . . . . 9  |-  ( 2nd `  ( Q `  b
) )  e.  _V
10 opelxpi 4737 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( 2nd `  ( Q `
 b ) )  e.  _V )  ->  <. b ,  ( 2nd `  ( Q `  b
) ) >.  e.  ( om  X.  _V )
)
119, 10mpan2 652 . . . . . . . 8  |-  ( b  e.  om  ->  <. b ,  ( 2nd `  ( Q `  b )
) >.  e.  ( om 
X.  _V ) )
128, 11eqeltrd 2370 . . . . . . 7  |-  ( b  e.  om  ->  (
( Q  |`  om ) `  b )  e.  ( om  X.  _V )
)
1312rgen 2621 . . . . . 6  |-  A. b  e.  om  ( ( Q  |`  om ) `  b
)  e.  ( om 
X.  _V )
14 ffnfv 5701 . . . . . 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 886 . . . . 5  |-  ( Q  |`  om ) : om --> ( om  X.  _V )
16 frn 5411 . . . . 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 4040 . . . . . . . . . 10  |-  ( a ran  ( Q  |`  om ) b  <->  <. a ,  b >.  e.  ran  ( Q  |`  om )
)
19 fvelrnb 5586 . . . . . . . . . . 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 5558 . . . . . . . . . . . 12  |-  ( c  e.  om  ->  (
( Q  |`  om ) `  c )  =  ( Q `  c ) )
2221eqeq1d 2304 . . . . . . . . . . 11  |-  ( c  e.  om  ->  (
( ( Q  |`  om ) `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  b >. )
)
2322rexbiia 2589 . . . . . . . . . 10  |-  ( E. c  e.  om  (
( Q  |`  om ) `  c )  =  <. a ,  b >.  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
2418, 20, 233bitri 262 . . . . . . . . 9  |-  ( a ran  ( Q  |`  om ) b  <->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
252seqomlem1 6478 . . . . . . . . . . . . . . . 16  |-  ( c  e.  om  ->  ( Q `  c )  =  <. c ,  ( 2nd `  ( Q `
 c ) )
>. )
2625adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( Q `  c
)  =  <. c ,  ( 2nd `  ( Q `  c )
) >. )
2726eqeq1d 2304 . . . . . . . . . . . . . 14  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  <->  <. c ,  ( 2nd `  ( Q `  c )
) >.  =  <. a ,  b >. )
)
28 vex 2804 . . . . . . . . . . . . . . 15  |-  c  e. 
_V
29 fvex 5555 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( Q `  c
) )  e.  _V
3028, 29opth1 4260 . . . . . . . . . . . . . 14  |-  ( <.
c ,  ( 2nd `  ( Q `  c
) ) >.  =  <. a ,  b >.  ->  c  =  a )
3127, 30syl6bi 219 . . . . . . . . . . . . 13  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  c  =  a ) )
32 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  ( Q `  c )  =  ( Q `  a ) )
3332eqeq1d 2304 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  <->  ( Q `  a )  =  <. a ,  b >. )
)
3433biimpd 198 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
3531, 34syli 33 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  ( Q `  a )  =  <. a ,  b
>. ) )
36 fveq2 5541 . . . . . . . . . . . . 13  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  ( 2nd `  ( Q `  a )
)  =  ( 2nd `  <. a ,  b
>. ) )
37 vex 2804 . . . . . . . . . . . . . 14  |-  a  e. 
_V
38 vex 2804 . . . . . . . . . . . . . 14  |-  b  e. 
_V
3937, 38op2nd 6145 . . . . . . . . . . . . 13  |-  ( 2nd `  <. a ,  b
>. )  =  b
4036, 39syl6req 2345 . . . . . . . . . . . 12  |-  ( ( Q `  a )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) )
4135, 40syl6 29 . . . . . . . . . . 11  |-  ( ( a  e.  om  /\  c  e.  om )  ->  ( ( Q `  c )  =  <. a ,  b >.  ->  b  =  ( 2nd `  ( Q `  a )
) ) )
4241rexlimdva 2680 . . . . . . . . . 10  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>.  ->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
432seqomlem1 6478 . . . . . . . . . . . 12  |-  ( a  e.  om  ->  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )
4432eqeq1d 2304 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4544rspcev 2897 . . . . . . . . . . . 12  |-  ( ( a  e.  om  /\  ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4643, 45mpdan 649 . . . . . . . . . . 11  |-  ( a  e.  om  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
47 opeq2 3813 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  <. a ,  b >.  =  <. a ,  ( 2nd `  ( Q `  a )
) >. )
4847eqeq2d 2307 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  ( Q `  a )
)  ->  ( ( Q `  c )  =  <. a ,  b
>. 
<->  ( Q `  c
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. ) )
4948rexbidv 2577 . . . . . . . . . . 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 213 . . . . . . . . . 10  |-  ( a  e.  om  ->  (
b  =  ( 2nd `  ( Q `  a
) )  ->  E. c  e.  om  ( Q `  c )  =  <. a ,  b >. )
)
5142, 50impbid 183 . . . . . . . . 9  |-  ( a  e.  om  ->  ( E. c  e.  om  ( Q `  c )  =  <. a ,  b
>. 
<->  b  =  ( 2nd `  ( Q `  a
) ) ) )
5224, 51syl5bb 248 . . . . . . . 8  |-  ( a  e.  om  ->  (
a ran  ( Q  |` 
om ) b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5352alrimiv 1621 . . . . . . 7  |-  ( a  e.  om  ->  A. b
( a ran  ( Q  |`  om ) b  <-> 
b  =  ( 2nd `  ( Q `  a
) ) ) )
54 fvex 5555 . . . . . . . 8  |-  ( 2nd `  ( Q `  a
) )  e.  _V
55 eqeq2 2305 . . . . . . . . . 10  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( b  =  c  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) )
5655bibi2d 309 . . . . . . . . 9  |-  ( c  =  ( 2nd `  ( Q `  a )
)  ->  ( (
a ran  ( Q  |` 
om ) b  <->  b  =  c )  <->  ( a ran  ( Q  |`  om )
b  <->  b  =  ( 2nd `  ( Q `
 a ) ) ) ) )
5756albidv 1615 . . . . . . . 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 2888 . . . . . . 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 15 . . . . . 6  |-  ( a  e.  om  ->  E. c A. b ( a ran  ( Q  |`  om )
b  <->  b  =  c ) )
60 df-eu 2160 . . . . . 6  |-  ( E! b  a ran  ( Q  |`  om ) b  <->  E. c A. b ( a ran  ( Q  |`  om ) b  <->  b  =  c ) )
6159, 60sylibr 203 . . . . 5  |-  ( a  e.  om  ->  E! b  a ran  ( Q  |`  om ) b )
6261rgen 2621 . . . 4  |-  A. a  e.  om  E! b  a ran  ( Q  |`  om ) b
63 dff3 5689 . . . 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 886 . . 3  |-  ran  ( Q  |`  om ) : om --> _V
65 df-ima 4718 . . . 4  |-  ( Q
" om )  =  ran  ( Q  |`  om )
6665feq1i 5399 . . 3  |-  ( ( Q " om ) : om --> _V  <->  ran  ( Q  |`  om ) : om --> _V )
6764, 66mpbir 200 . 2  |-  ( Q
" om ) : om --> _V
68 dffn2 5406 . 2  |-  ( ( Q " om )  Fn  om  <->  ( Q " om ) : om --> _V )
6967, 68mpbir 200 1  |-  ( Q
" om )  Fn 
om
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   <.cop 3656   class class class wbr 4039    _I cid 4320   suc csuc 4410   omcom 4672    X. cxp 4703   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   reccrdg 6438
This theorem is referenced by:  seqomlem3  6480  seqomlem4  6481  fnseqom  6483
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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