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

Theorem cantnfvalf 8080
Description: Lemma for cantnf 8108. The function appearing in cantnfval 8083 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
Hypothesis
Ref Expression
cantnfvalf.f  |-  F  = seq𝜔 ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D
) ) ,  (/) )
Assertion
Ref Expression
cantnfvalf  |-  F : om
--> On
Distinct variable groups:    z, k, A    B, k, z
Allowed substitution hints:    C( z, k)    D( z, k)    F( z, k)

Proof of Theorem cantnfvalf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfvalf.f . . 3  |-  F  = seq𝜔 ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D
) ) ,  (/) )
21fnseqom 7117 . 2  |-  F  Fn  om
3 nn0suc 6702 . . . 4  |-  ( x  e.  om  ->  (
x  =  (/)  \/  E. y  e.  om  x  =  suc  y ) )
4 fveq2 5864 . . . . . . 7  |-  ( x  =  (/)  ->  ( F `
 x )  =  ( F `  (/) ) )
5 0ex 4577 . . . . . . . 8  |-  (/)  e.  _V
61seqom0g 7118 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( F `  (/) )  =  (/) )
75, 6ax-mp 5 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
84, 7syl6eq 2524 . . . . . 6  |-  ( x  =  (/)  ->  ( F `
 x )  =  (/) )
9 0elon 4931 . . . . . 6  |-  (/)  e.  On
108, 9syl6eqel 2563 . . . . 5  |-  ( x  =  (/)  ->  ( F `
 x )  e.  On )
111seqomsuc 7119 . . . . . . . . 9  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( y ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) ( F `  y ) ) )
12 df-ov 6285 . . . . . . . . 9  |-  ( y ( k  e.  A ,  z  e.  B  |->  ( C  +o  D
) ) ( F `
 y ) )  =  ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) `
 <. y ,  ( F `  y )
>. )
1311, 12syl6eq 2524 . . . . . . . 8  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) `
 <. y ,  ( F `  y )
>. ) )
14 df-ov 6285 . . . . . . . . . . . 12  |-  ( C  +o  D )  =  (  +o  `  <. C ,  D >. )
15 fnoa 7155 . . . . . . . . . . . . . 14  |-  +o  Fn  ( On  X.  On )
16 oacl 7182 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  +o  y
)  e.  On )
1716rgen2a 2891 . . . . . . . . . . . . . 14  |-  A. x  e.  On  A. y  e.  On  ( x  +o  y )  e.  On
18 ffnov 6388 . . . . . . . . . . . . . 14  |-  (  +o  : ( On  X.  On ) --> On  <->  (  +o  Fn  ( On  X.  On )  /\  A. x  e.  On  A. y  e.  On  ( x  +o  y )  e.  On ) )
1915, 17, 18mpbir2an 918 . . . . . . . . . . . . 13  |-  +o  :
( On  X.  On )
--> On
2019, 9f0cli 6030 . . . . . . . . . . . 12  |-  (  +o 
`  <. C ,  D >. )  e.  On
2114, 20eqeltri 2551 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
2221rgen2w 2826 . . . . . . . . . 10  |-  A. k  e.  A  A. z  e.  B  ( C  +o  D )  e.  On
23 eqid 2467 . . . . . . . . . . 11  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )  =  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )
2423fmpt2 6848 . . . . . . . . . 10  |-  ( A. k  e.  A  A. z  e.  B  ( C  +o  D )  e.  On  <->  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) : ( A  X.  B
) --> On )
2522, 24mpbi 208 . . . . . . . . 9  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) : ( A  X.  B ) --> On
2625, 9f0cli 6030 . . . . . . . 8  |-  ( ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) `  <. y ,  ( F `  y ) >. )  e.  On
2713, 26syl6eqel 2563 . . . . . . 7  |-  ( y  e.  om  ->  ( F `  suc  y )  e.  On )
28 fveq2 5864 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( F `  x
)  =  ( F `
 suc  y )
)
2928eleq1d 2536 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( F `  x )  e.  On  <->  ( F `  suc  y
)  e.  On ) )
3027, 29syl5ibrcom 222 . . . . . 6  |-  ( y  e.  om  ->  (
x  =  suc  y  ->  ( F `  x
)  e.  On ) )
3130rexlimiv 2949 . . . . 5  |-  ( E. y  e.  om  x  =  suc  y  ->  ( F `  x )  e.  On )
3210, 31jaoi 379 . . . 4  |-  ( ( x  =  (/)  \/  E. y  e.  om  x  =  suc  y )  -> 
( F `  x
)  e.  On )
333, 32syl 16 . . 3  |-  ( x  e.  om  ->  ( F `  x )  e.  On )
3433rgen 2824 . 2  |-  A. x  e.  om  ( F `  x )  e.  On
35 ffnfv 6045 . 2  |-  ( F : om --> On  <->  ( F  Fn  om  /\  A. x  e.  om  ( F `  x )  e.  On ) )
362, 34, 35mpbir2an 918 1  |-  F : om
--> On
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   (/)c0 3785   <.cop 4033   Oncon0 4878   suc csuc 4880    X. cxp 4997    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   omcom 6678  seq𝜔cseqom 7109    +o coa 7124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-seqom 7110  df-oadd 7131
This theorem is referenced by:  cantnfval2  8084  cantnfle  8086  cantnflt  8087  cantnflem1d  8103  cantnflem1  8104  cantnfval2OLD  8114  cantnfleOLD  8116  cantnfltOLD  8117  cantnflem1dOLD  8126  cantnflem1OLD  8127  cnfcomlem  8139  cnfcomlemOLD  8147
  Copyright terms: Public domain W3C validator