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Theorem cantnfvalf 7366
Description: Lemma for cantnf 7395. The function appearing in cantnfval 7369 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 6467 . 2  |-  F  Fn  om
3 nn0suc 4680 . . . 4  |-  ( x  e.  om  ->  (
x  =  (/)  \/  E. y  e.  om  x  =  suc  y ) )
4 fveq2 5525 . . . . . . 7  |-  ( x  =  (/)  ->  ( F `
 x )  =  ( F `  (/) ) )
5 0ex 4150 . . . . . . . 8  |-  (/)  e.  _V
61seqom0g 6468 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( F `  (/) )  =  (/) )
75, 6ax-mp 8 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
84, 7syl6eq 2331 . . . . . 6  |-  ( x  =  (/)  ->  ( F `
 x )  =  (/) )
9 0elon 4445 . . . . . 6  |-  (/)  e.  On
108, 9syl6eqel 2371 . . . . 5  |-  ( x  =  (/)  ->  ( F `
 x )  e.  On )
111seqomsuc 6469 . . . . . . . . 9  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( y ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) ( F `  y ) ) )
12 df-ov 5861 . . . . . . . . 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 2331 . . . . . . . 8  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) `
 <. y ,  ( F `  y )
>. ) )
14 df-ov 5861 . . . . . . . . . . . 12  |-  ( C  +o  D )  =  (  +o  `  <. C ,  D >. )
15 fnoa 6507 . . . . . . . . . . . . . 14  |-  +o  Fn  ( On  X.  On )
16 oacl 6534 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  +o  y
)  e.  On )
1716rgen2a 2609 . . . . . . . . . . . . . 14  |-  A. x  e.  On  A. y  e.  On  ( x  +o  y )  e.  On
18 ffnov 5948 . . . . . . . . . . . . . 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 886 . . . . . . . . . . . . 13  |-  +o  :
( On  X.  On )
--> On
2019, 9f0cli 5671 . . . . . . . . . . . 12  |-  (  +o 
`  <. C ,  D >. )  e.  On
2114, 20eqeltri 2353 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
2221rgen2w 2611 . . . . . . . . . 10  |-  A. k  e.  A  A. z  e.  B  ( C  +o  D )  e.  On
23 eqid 2283 . . . . . . . . . . 11  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )  =  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )
2423fmpt2 6191 . . . . . . . . . 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 199 . . . . . . . . 9  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) : ( A  X.  B ) --> On
2625, 9f0cli 5671 . . . . . . . 8  |-  ( ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) `  <. y ,  ( F `  y ) >. )  e.  On
2713, 26syl6eqel 2371 . . . . . . 7  |-  ( y  e.  om  ->  ( F `  suc  y )  e.  On )
28 fveq2 5525 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( F `  x
)  =  ( F `
 suc  y )
)
2928eleq1d 2349 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( F `  x )  e.  On  <->  ( F `  suc  y
)  e.  On ) )
3027, 29syl5ibrcom 213 . . . . . 6  |-  ( y  e.  om  ->  (
x  =  suc  y  ->  ( F `  x
)  e.  On ) )
3130rexlimiv 2661 . . . . 5  |-  ( E. y  e.  om  x  =  suc  y  ->  ( F `  x )  e.  On )
3210, 31jaoi 368 . . . 4  |-  ( ( x  =  (/)  \/  E. y  e.  om  x  =  suc  y )  -> 
( F `  x
)  e.  On )
333, 32syl 15 . . 3  |-  ( x  e.  om  ->  ( F `  x )  e.  On )
3433rgen 2608 . 2  |-  A. x  e.  om  ( F `  x )  e.  On
35 ffnfv 5685 . 2  |-  ( F : om --> On  <->  ( F  Fn  om  /\  A. x  e.  om  ( F `  x )  e.  On ) )
362, 34, 35mpbir2an 886 1  |-  F : om
--> On
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   (/)c0 3455   <.cop 3643   Oncon0 4392   suc csuc 4394   omcom 4656    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459    +o coa 6476
This theorem is referenced by:  cantnfval2  7370  cantnfle  7372  cantnflt  7373  cantnflem1d  7390  cantnflem1  7391  cnfcomlem  7402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-seqom 6460  df-oadd 6483
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