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Theorem cantnfvalf 8118
Description: Lemma for cantnf 8146. The function appearing in cantnfval 8121 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 7159 . 2  |-  F  Fn  om
3 nn0suc 6710 . . . 4  |-  ( x  e.  om  ->  (
x  =  (/)  \/  E. y  e.  om  x  =  suc  y ) )
4 fveq2 5851 . . . . . . 7  |-  ( x  =  (/)  ->  ( F `
 x )  =  ( F `  (/) ) )
5 0ex 4528 . . . . . . . 8  |-  (/)  e.  _V
61seqom0g 7160 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( F `  (/) )  =  (/) )
75, 6ax-mp 5 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
84, 7syl6eq 2461 . . . . . 6  |-  ( x  =  (/)  ->  ( F `
 x )  =  (/) )
9 0elon 5465 . . . . . 6  |-  (/)  e.  On
108, 9syl6eqel 2500 . . . . 5  |-  ( x  =  (/)  ->  ( F `
 x )  e.  On )
111seqomsuc 7161 . . . . . . . . 9  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( y ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) ( F `  y ) ) )
12 df-ov 6283 . . . . . . . . 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 2461 . . . . . . . 8  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) `
 <. y ,  ( F `  y )
>. ) )
14 df-ov 6283 . . . . . . . . . . . 12  |-  ( C  +o  D )  =  (  +o  `  <. C ,  D >. )
15 fnoa 7197 . . . . . . . . . . . . . 14  |-  +o  Fn  ( On  X.  On )
16 oacl 7224 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  +o  y
)  e.  On )
1716rgen2a 2833 . . . . . . . . . . . . . 14  |-  A. x  e.  On  A. y  e.  On  ( x  +o  y )  e.  On
18 ffnov 6389 . . . . . . . . . . . . . 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 923 . . . . . . . . . . . . 13  |-  +o  :
( On  X.  On )
--> On
2019, 9f0cli 6022 . . . . . . . . . . . 12  |-  (  +o 
`  <. C ,  D >. )  e.  On
2114, 20eqeltri 2488 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
2221rgen2w 2768 . . . . . . . . . 10  |-  A. k  e.  A  A. z  e.  B  ( C  +o  D )  e.  On
23 eqid 2404 . . . . . . . . . . 11  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )  =  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )
2423fmpt2 6853 . . . . . . . . . 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 210 . . . . . . . . 9  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) : ( A  X.  B ) --> On
2625, 9f0cli 6022 . . . . . . . 8  |-  ( ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) `  <. y ,  ( F `  y ) >. )  e.  On
2713, 26syl6eqel 2500 . . . . . . 7  |-  ( y  e.  om  ->  ( F `  suc  y )  e.  On )
28 fveq2 5851 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( F `  x
)  =  ( F `
 suc  y )
)
2928eleq1d 2473 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( F `  x )  e.  On  <->  ( F `  suc  y
)  e.  On ) )
3027, 29syl5ibrcom 224 . . . . . 6  |-  ( y  e.  om  ->  (
x  =  suc  y  ->  ( F `  x
)  e.  On ) )
3130rexlimiv 2892 . . . . 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 17 . . 3  |-  ( x  e.  om  ->  ( F `  x )  e.  On )
3433rgen 2766 . 2  |-  A. x  e.  om  ( F `  x )  e.  On
35 ffnfv 6038 . 2  |-  ( F : om --> On  <->  ( F  Fn  om  /\  A. x  e.  om  ( F `  x )  e.  On ) )
362, 34, 35mpbir2an 923 1  |-  F : om
--> On
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   _Vcvv 3061   (/)c0 3740   <.cop 3980    X. cxp 4823   Oncon0 5412   suc csuc 5414    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   omcom 6685  seq𝜔cseqom 7151    +o coa 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-seqom 7152  df-oadd 7173
This theorem is referenced by:  cantnfval2  8122  cantnfle  8124  cantnflt  8125  cantnflem1d  8141  cantnflem1  8142  cantnfval2OLD  8152  cantnfleOLD  8154  cantnfltOLD  8155  cantnflem1dOLD  8164  cantnflem1OLD  8165  cnfcomlem  8177  cnfcomlemOLD  8185
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