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Theorem cantnfvalf 8101
Description: Lemma for cantnf 8129. The function appearing in cantnfval 8104 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 7138 . 2  |-  F  Fn  om
3 nn0suc 6723 . . . 4  |-  ( x  e.  om  ->  (
x  =  (/)  \/  E. y  e.  om  x  =  suc  y ) )
4 fveq2 5872 . . . . . . 7  |-  ( x  =  (/)  ->  ( F `
 x )  =  ( F `  (/) ) )
5 0ex 4587 . . . . . . . 8  |-  (/)  e.  _V
61seqom0g 7139 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( F `  (/) )  =  (/) )
75, 6ax-mp 5 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
84, 7syl6eq 2514 . . . . . 6  |-  ( x  =  (/)  ->  ( F `
 x )  =  (/) )
9 0elon 4940 . . . . . 6  |-  (/)  e.  On
108, 9syl6eqel 2553 . . . . 5  |-  ( x  =  (/)  ->  ( F `
 x )  e.  On )
111seqomsuc 7140 . . . . . . . . 9  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( y ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) ( F `  y ) ) )
12 df-ov 6299 . . . . . . . . 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 2514 . . . . . . . 8  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) `
 <. y ,  ( F `  y )
>. ) )
14 df-ov 6299 . . . . . . . . . . . 12  |-  ( C  +o  D )  =  (  +o  `  <. C ,  D >. )
15 fnoa 7176 . . . . . . . . . . . . . 14  |-  +o  Fn  ( On  X.  On )
16 oacl 7203 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  +o  y
)  e.  On )
1716rgen2a 2884 . . . . . . . . . . . . . 14  |-  A. x  e.  On  A. y  e.  On  ( x  +o  y )  e.  On
18 ffnov 6405 . . . . . . . . . . . . . 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 920 . . . . . . . . . . . . 13  |-  +o  :
( On  X.  On )
--> On
2019, 9f0cli 6043 . . . . . . . . . . . 12  |-  (  +o 
`  <. C ,  D >. )  e.  On
2114, 20eqeltri 2541 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
2221rgen2w 2819 . . . . . . . . . 10  |-  A. k  e.  A  A. z  e.  B  ( C  +o  D )  e.  On
23 eqid 2457 . . . . . . . . . . 11  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )  =  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )
2423fmpt2 6866 . . . . . . . . . 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 6043 . . . . . . . 8  |-  ( ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) `  <. y ,  ( F `  y ) >. )  e.  On
2713, 26syl6eqel 2553 . . . . . . 7  |-  ( y  e.  om  ->  ( F `  suc  y )  e.  On )
28 fveq2 5872 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( F `  x
)  =  ( F `
 suc  y )
)
2928eleq1d 2526 . . . . . . 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 2943 . . . . 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 2817 . 2  |-  A. x  e.  om  ( F `  x )  e.  On
35 ffnfv 6058 . 2  |-  ( F : om --> On  <->  ( F  Fn  om  /\  A. x  e.  om  ( F `  x )  e.  On ) )
362, 34, 35mpbir2an 920 1  |-  F : om
--> On
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109   (/)c0 3793   <.cop 4038   Oncon0 4887   suc csuc 4889    X. cxp 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   omcom 6699  seq𝜔cseqom 7130    +o coa 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-seqom 7131  df-oadd 7152
This theorem is referenced by:  cantnfval2  8105  cantnfle  8107  cantnflt  8108  cantnflem1d  8124  cantnflem1  8125  cantnfval2OLD  8135  cantnfleOLD  8137  cantnfltOLD  8138  cantnflem1dOLD  8147  cantnflem1OLD  8148  cnfcomlem  8160  cnfcomlemOLD  8168
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