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Theorem cnfcom2lemOLD 8185
Description: Lemma for cnfcom2OLD 8186. (Contributed by Mario Carneiro, 30-May-2015.) Obsolete version of cnfcom2lem 8177 as of 3-Jul-2019. (New usage is discouraged.) (Proof modification 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 3743 . . . . . 6  |-  ( (/)  e.  B  ->  -.  B  =  (/) )
31, 2syl 17 . . . . 5  |-  ( ph  ->  -.  B  =  (/) )
4 cnfcomOLD.f . . . . . . . . . . . . . 14  |-  F  =  ( `' ( om CNF 
A ) `  B
)
5 cnfcomOLD.s . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( om CNF  A
)
6 omelon 8096 . . . . . . . . . . . . . . . . . 18  |-  om  e.  On
76a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  om  e.  On )
8 cnfcomOLD.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
95, 7, 8cantnff1o 8169 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
) )
10 f1ocnv 5811 . . . . . . . . . . . . . . . 16  |-  ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  ->  `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S )
11 f1of 5799 . . . . . . . . . . . . . . . 16  |-  ( `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S  ->  `' ( om CNF  A ) : ( om  ^o  A ) --> S )
129, 10, 113syl 18 . . . . . . . . . . . . . . 15  |-  ( ph  ->  `' ( om CNF  A
) : ( om 
^o  A ) --> S )
13 cnfcomOLD.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  ( om 
^o  A ) )
1412, 13ffvelrnd 6010 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( om CNF 
A ) `  B
)  e.  S )
154, 14syl5eqel 2494 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  S )
165, 7, 8cantnfsOLD 8147 . . . . . . . . . . . . 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 457 . . . . . . . . . . 11  |-  ( ph  ->  F : A --> om )
1918adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F : A
--> om )
2019feqmptd 5902 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21 dif0 3842 . . . . . . . . . . . 12  |-  ( A 
\  (/) )  =  A
2221eleq2i 2480 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  (/) )  <->  x  e.  A
)
23 df1o2 7179 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
2423difeq2i 3558 . . . . . . . . . . . . . . 15  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2524imaeq2i 5155 . . . . . . . . . . . . . 14  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
26 simpr 459 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  =  (/) )
27 cnvimass 5177 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
28 fdm 5718 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : A --> om  ->  dom 
F  =  A )
2918, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  F  =  A )
3027, 29syl5sseq 3490 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  A
)
318, 30ssexd 4541 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
32 cnfcomOLD.g . . . . . . . . . . . . . . . . . . . . 21  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
335, 7, 8, 32, 15cantnfclOLD 8148 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
3433simpld 457 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
3532oien 7997 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3631, 34, 35syl2anc 659 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3736adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  ~~  ( `' F "
( _V  \  1o ) ) )
3826, 37eqbrtrrd 4417 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  dom  G  =  (/) )  ->  (/)  ~~  ( `' F " ( _V 
\  1o ) ) )
3938ensymd 7604 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
40 en0 7616 . . . . . . . . . . . . . . 15  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
4139, 40sylib 196 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) )  =  (/) )
4225, 41syl5eqr 2457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  =  (/) )
43 ss0b 3769 . . . . . . . . . . . . 13  |-  ( ( `' F " ( _V 
\  { (/) } ) )  C_  (/)  <->  ( `' F " ( _V  \  { (/) } ) )  =  (/) )
4442, 43sylibr 212 . . . . . . . . . . . 12  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  C_  (/) )
4519, 44suppssrOLD 5999 . . . . . . . . . . 11  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  ( A  \  (/) ) )  ->  ( F `  x )  =  (/) )
4622, 45sylan2br 474 . . . . . . . . . 10  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  (/) )
4746mpteq2dva 4481 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  (/) ) )
4820, 47eqtrd 2443 . . . . . . . 8  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  (/) ) )
49 fconstmpt 4867 . . . . . . . 8  |-  ( A  X.  { (/) } )  =  ( x  e.  A  |->  (/) )
5048, 49syl6eqr 2461 . . . . . . 7  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( A  X.  { (/)
} ) )
5150fveq2d 5853 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( A  X.  { (/)
} ) ) )
524fveq2i 5852 . . . . . . . 8  |-  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( `' ( om CNF 
A ) `  B
) )
53 f1ocnvfv2 6164 . . . . . . . . 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 659 . . . . . . . 8  |-  ( ph  ->  ( ( om CNF  A
) `  ( `' ( om CNF  A ) `  B ) )  =  B )
5552, 54syl5eq 2455 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  F )  =  B )
5655adantr 463 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  B )
57 peano1 6703 . . . . . . . . 9  |-  (/)  e.  om
5857a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  om )
595, 7, 8, 58cantnf0 8126 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  ( A  X.  { (/) } ) )  =  (/) )
6059adantr 463 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  ( A  X.  { (/) } ) )  =  (/) )
6151, 56, 603eqtr3d 2451 . . . . 5  |-  ( (
ph  /\  dom  G  =  (/) )  ->  B  =  (/) )
623, 61mtand 657 . . . 4  |-  ( ph  ->  -.  dom  G  =  (/) )
6333simprd 461 . . . . 5  |-  ( ph  ->  dom  G  e.  om )
64 nnlim 6696 . . . . 5  |-  ( dom 
G  e.  om  ->  -. 
Lim  dom  G )
6563, 64syl 17 . . . 4  |-  ( ph  ->  -.  Lim  dom  G
)
66 ioran 488 . . . 4  |-  ( -.  ( dom  G  =  (/)  \/  Lim  dom  G
)  <->  ( -.  dom  G  =  (/)  /\  -.  Lim  dom 
G ) )
6762, 65, 66sylanbrc 662 . . 3  |-  ( ph  ->  -.  ( dom  G  =  (/)  \/  Lim  dom  G ) )
6832oicl 7988 . . . 4  |-  Ord  dom  G
69 unizlim 5526 . . . 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 303 . 2  |-  ( ph  ->  -.  dom  G  = 
U. dom  G )
72 orduniorsuc 6648 . . . 4  |-  ( Ord 
dom  G  ->  ( dom 
G  =  U. dom  G  \/  dom  G  =  suc  U. dom  G
) )
7368, 72mp1i 13 . . 3  |-  ( ph  ->  ( dom  G  = 
U. dom  G  \/  dom  G  =  suc  U. dom  G ) )
7473ord 375 . 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 366    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    \ cdif 3411    u. cun 3412    C_ wss 3414   (/)c0 3738   {csn 3972   U.cuni 4191   class class class wbr 4395    |-> cmpt 4453    _E cep 4732    We wwe 4781    X. cxp 4821   `'ccnv 4822   dom cdm 4823   "cima 4826   Ord word 5409   Oncon0 5410   Lim wlim 5411   suc csuc 5412   -->wf 5565   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   omcom 6683  seq𝜔cseqom 7149   1oc1o 7160    +o coa 7164    .o comu 7165    ^o coe 7166    ~~ cen 7551   Fincfn 7554  OrdIsocoi 7968   CNF ccnf 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-seqom 7150  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-oexp 7173  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-oi 7969  df-cnf 8111
This theorem is referenced by:  cnfcom2OLD  8186  cnfcom3lemOLD  8187  cnfcom3OLD  8188
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