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Theorem cnfcom2lemOLD 7934
Description: Lemma for cnfcom2OLD 7935. (Contributed by Mario Carneiro, 30-May-2015.) Obsolete version of cnfcom2lem 7926 as of 3-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cnfcomOLD.s  |-  S  =  dom  ( om CNF  A
)
cnfcomOLD.a  |-  ( ph  ->  A  e.  On )
cnfcomOLD.b  |-  ( ph  ->  B  e.  ( om 
^o  A ) )
cnfcomOLD.f  |-  F  =  ( `' ( om CNF 
A ) `  B
)
cnfcomOLD.g  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
cnfcomOLD.h  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( M  +o  z
) ) ,  (/) )
cnfcomOLD.t  |-  T  = seq𝜔 ( ( k  e.  _V ,  f  e.  _V  |->  K ) ,  (/) )
cnfcomOLD.m  |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `
 ( G `  k ) ) )
cnfcomOLD.k  |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( M  +o  x ) ) )
cnfcomOLD.w  |-  W  =  ( G `  U. dom  G )
cnfcom2OLD.1  |-  ( ph  -> 
(/)  e.  B )
Assertion
Ref Expression
cnfcom2lemOLD  |-  ( ph  ->  dom  G  =  suc  U.
dom  G )
Distinct variable groups:    x, k,
z, A    x, M    f, k, x, z, F   
z, T    x, W    f, G, k, x, z   
f, H, x    S, k, z    ph, k, x, z
Allowed substitution hints:    ph( f)    A( f)    B( x, z, f, k)    S( x, f)    T( x, f, k)    H( z, k)    K( x, z, f, k)    M( z, f, k)    W( z, f, k)

Proof of Theorem cnfcom2lemOLD
StepHypRef Expression
1 cnfcom2OLD.1 . . . . . 6  |-  ( ph  -> 
(/)  e.  B )
2 n0i 3637 . . . . . 6  |-  ( (/)  e.  B  ->  -.  B  =  (/) )
31, 2syl 16 . . . . 5  |-  ( ph  ->  -.  B  =  (/) )
4 cnfcomOLD.f . . . . . . . . . . . . . 14  |-  F  =  ( `' ( om CNF 
A ) `  B
)
5 cnfcomOLD.s . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( om CNF  A
)
6 omelon 7844 . . . . . . . . . . . . . . . . . 18  |-  om  e.  On
76a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  om  e.  On )
8 cnfcomOLD.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
95, 7, 8cantnff1o 7918 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
) )
10 f1ocnv 5648 . . . . . . . . . . . . . . . 16  |-  ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  ->  `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S )
11 f1of 5636 . . . . . . . . . . . . . . . 16  |-  ( `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S  ->  `' ( om CNF  A ) : ( om  ^o  A ) --> S )
129, 10, 113syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  `' ( om CNF  A
) : ( om 
^o  A ) --> S )
13 cnfcomOLD.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  ( om 
^o  A ) )
1412, 13ffvelrnd 5839 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( om CNF 
A ) `  B
)  e.  S )
154, 14syl5eqel 2522 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  S )
165, 7, 8cantnfsOLD 7896 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  e.  S  <->  ( F : A --> om  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
1715, 16mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( F : A --> om  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
1817simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  F : A --> om )
1918adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F : A
--> om )
2019feqmptd 5739 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21 dif0 3744 . . . . . . . . . . . 12  |-  ( A 
\  (/) )  =  A
2221eleq2i 2502 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  (/) )  <->  x  e.  A
)
23 df1o2 6924 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
2423difeq2i 3466 . . . . . . . . . . . . . . 15  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2524imaeq2i 5162 . . . . . . . . . . . . . 14  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
26 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  =  (/) )
27 cnvimass 5184 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
28 fdm 5558 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : A --> om  ->  dom 
F  =  A )
2918, 28syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  F  =  A )
3027, 29syl5sseq 3399 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  A
)
318, 30ssexd 4434 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
32 cnfcomOLD.g . . . . . . . . . . . . . . . . . . . . 21  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
335, 7, 8, 32, 15cantnfclOLD 7897 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
3433simpld 459 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
3532oien 7744 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3631, 34, 35syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3736adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  ~~  ( `' F "
( _V  \  1o ) ) )
3826, 37eqbrtrrd 4309 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  dom  G  =  (/) )  ->  (/)  ~~  ( `' F " ( _V 
\  1o ) ) )
3938ensymd 7352 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
40 en0 7364 . . . . . . . . . . . . . . 15  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
4139, 40sylib 196 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) )  =  (/) )
4225, 41syl5eqr 2484 . . . . . . . . . . . . 13  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  =  (/) )
43 ss0b 3662 . . . . . . . . . . . . 13  |-  ( ( `' F " ( _V 
\  { (/) } ) )  C_  (/)  <->  ( `' F " ( _V  \  { (/) } ) )  =  (/) )
4442, 43sylibr 212 . . . . . . . . . . . 12  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  C_  (/) )
4519, 44suppssrOLD 5832 . . . . . . . . . . 11  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  ( A  \  (/) ) )  ->  ( F `  x )  =  (/) )
4622, 45sylan2br 476 . . . . . . . . . 10  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  (/) )
4746mpteq2dva 4373 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  (/) ) )
4820, 47eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  (/) ) )
49 fconstmpt 4877 . . . . . . . 8  |-  ( A  X.  { (/) } )  =  ( x  e.  A  |->  (/) )
5048, 49syl6eqr 2488 . . . . . . 7  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( A  X.  { (/)
} ) )
5150fveq2d 5690 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( A  X.  { (/)
} ) ) )
524fveq2i 5689 . . . . . . . 8  |-  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( `' ( om CNF 
A ) `  B
) )
53 f1ocnvfv2 5979 . . . . . . . . 9  |-  ( ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  /\  B  e.  ( om  ^o  A ) )  ->  ( ( om CNF  A ) `  ( `' ( om CNF  A
) `  B )
)  =  B )
549, 13, 53syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( om CNF  A
) `  ( `' ( om CNF  A ) `  B ) )  =  B )
5552, 54syl5eq 2482 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  F )  =  B )
5655adantr 465 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  B )
57 peano1 6490 . . . . . . . . 9  |-  (/)  e.  om
5857a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  om )
595, 7, 8, 58cantnf0 7875 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  ( A  X.  { (/) } ) )  =  (/) )
6059adantr 465 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  ( A  X.  { (/) } ) )  =  (/) )
6151, 56, 603eqtr3d 2478 . . . . 5  |-  ( (
ph  /\  dom  G  =  (/) )  ->  B  =  (/) )
623, 61mtand 659 . . . 4  |-  ( ph  ->  -.  dom  G  =  (/) )
6333simprd 463 . . . . 5  |-  ( ph  ->  dom  G  e.  om )
64 nnlim 6484 . . . . 5  |-  ( dom 
G  e.  om  ->  -. 
Lim  dom  G )
6563, 64syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  G
)
66 ioran 490 . . . 4  |-  ( -.  ( dom  G  =  (/)  \/  Lim  dom  G
)  <->  ( -.  dom  G  =  (/)  /\  -.  Lim  dom 
G ) )
6762, 65, 66sylanbrc 664 . . 3  |-  ( ph  ->  -.  ( dom  G  =  (/)  \/  Lim  dom  G ) )
6832oicl 7735 . . . 4  |-  Ord  dom  G
69 unizlim 4830 . . . 4  |-  ( Ord 
dom  G  ->  ( dom 
G  =  U. dom  G  <-> 
( dom  G  =  (/) 
\/  Lim  dom  G ) ) )
7068, 69ax-mp 5 . . 3  |-  ( dom 
G  =  U. dom  G  <-> 
( dom  G  =  (/) 
\/  Lim  dom  G ) )
7167, 70sylnibr 305 . 2  |-  ( ph  ->  -.  dom  G  = 
U. dom  G )
72 orduniorsuc 6436 . . . 4  |-  ( Ord 
dom  G  ->  ( dom 
G  =  U. dom  G  \/  dom  G  =  suc  U. dom  G
) )
7368, 72mp1i 12 . . 3  |-  ( ph  ->  ( dom  G  = 
U. dom  G  \/  dom  G  =  suc  U. dom  G ) )
7473ord 377 . 2  |-  ( ph  ->  ( -.  dom  G  =  U. dom  G  ->  dom  G  =  suc  U. dom  G ) )
7571, 74mpd 15 1  |-  ( ph  ->  dom  G  =  suc  U.
dom  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    \ cdif 3320    u. cun 3321    C_ wss 3323   (/)c0 3632   {csn 3872   U.cuni 4086   class class class wbr 4287    e. cmpt 4345    _E cep 4625    We wwe 4673   Ord word 4713   Oncon0 4714   Lim wlim 4715   suc csuc 4716    X. cxp 4833   `'ccnv 4834   dom cdm 4835   "cima 4838   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   omcom 6471  seq𝜔cseqom 6894   1oc1o 6905    +o coa 6909    .o comu 6910    ^o coe 6911    ~~ cen 7299   Fincfn 7302  OrdIsocoi 7715   CNF ccnf 7859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-seqom 6895  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-oexp 6918  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-oi 7716  df-cnf 7860
This theorem is referenced by:  cnfcom2OLD  7935  cnfcom3lemOLD  7936  cnfcom3OLD  7937
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