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Theorem cantnfvalf 7382
Description: Lemma for cantnf 7411. The function appearing in cantnfval 7385 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 6483 . 2  |-  F  Fn  om
3 nn0suc 4696 . . . 4  |-  ( x  e.  om  ->  (
x  =  (/)  \/  E. y  e.  om  x  =  suc  y ) )
4 fveq2 5541 . . . . . . 7  |-  ( x  =  (/)  ->  ( F `
 x )  =  ( F `  (/) ) )
5 0ex 4166 . . . . . . . 8  |-  (/)  e.  _V
61seqom0g 6484 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( F `  (/) )  =  (/) )
75, 6ax-mp 8 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
84, 7syl6eq 2344 . . . . . 6  |-  ( x  =  (/)  ->  ( F `
 x )  =  (/) )
9 0elon 4461 . . . . . 6  |-  (/)  e.  On
108, 9syl6eqel 2384 . . . . 5  |-  ( x  =  (/)  ->  ( F `
 x )  e.  On )
111seqomsuc 6485 . . . . . . . . 9  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( y ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) ( F `  y ) ) )
12 df-ov 5877 . . . . . . . . 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 2344 . . . . . . . 8  |-  ( y  e.  om  ->  ( F `  suc  y )  =  ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) `
 <. y ,  ( F `  y )
>. ) )
14 df-ov 5877 . . . . . . . . . . . 12  |-  ( C  +o  D )  =  (  +o  `  <. C ,  D >. )
15 fnoa 6523 . . . . . . . . . . . . . 14  |-  +o  Fn  ( On  X.  On )
16 oacl 6550 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  +o  y
)  e.  On )
1716rgen2a 2622 . . . . . . . . . . . . . 14  |-  A. x  e.  On  A. y  e.  On  ( x  +o  y )  e.  On
18 ffnov 5964 . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . 12  |-  (  +o 
`  <. C ,  D >. )  e.  On
2114, 20eqeltri 2366 . . . . . . . . . . 11  |-  ( C  +o  D )  e.  On
2221rgen2w 2624 . . . . . . . . . 10  |-  A. k  e.  A  A. z  e.  B  ( C  +o  D )  e.  On
23 eqid 2296 . . . . . . . . . . 11  |-  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )  =  ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) )
2423fmpt2 6207 . . . . . . . . . 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 5687 . . . . . . . 8  |-  ( ( k  e.  A , 
z  e.  B  |->  ( C  +o  D ) ) `  <. y ,  ( F `  y ) >. )  e.  On
2713, 26syl6eqel 2384 . . . . . . 7  |-  ( y  e.  om  ->  ( F `  suc  y )  e.  On )
28 fveq2 5541 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( F `  x
)  =  ( F `
 suc  y )
)
2928eleq1d 2362 . . . . . . 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 2674 . . . . 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 2621 . 2  |-  A. x  e.  om  ( F `  x )  e.  On
35 ffnfv 5701 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   (/)c0 3468   <.cop 3656   Oncon0 4408   suc csuc 4410   omcom 4672    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876  seq𝜔cseqom 6475    +o coa 6492
This theorem is referenced by:  cantnfval2  7386  cantnfle  7388  cantnflt  7389  cantnflem1d  7406  cantnflem1  7407  cnfcomlem  7418
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-rep 4147  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-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-seqom 6476  df-oadd 6499
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