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Theorem cnfcom2lem 7404
Description: Lemma for cnfcom2 7405. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
cnfcom.s  |-  S  =  dom  ( om CNF  A
)
cnfcom.a  |-  ( ph  ->  A  e.  On )
cnfcom.b  |-  ( ph  ->  B  e.  ( om 
^o  A ) )
cnfcom.f  |-  F  =  ( `' ( om CNF 
A ) `  B
)
cnfcom.g  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
cnfcom.h  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( M  +o  z
) ) ,  (/) )
cnfcom.t  |-  T  = seq𝜔 ( ( k  e.  _V ,  f  e.  _V  |->  K ) ,  (/) )
cnfcom.m  |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `
 ( G `  k ) ) )
cnfcom.k  |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( M  +o  x ) ) )
cnfcom.w  |-  W  =  ( G `  U. dom  G )
cnfcom2.1  |-  ( ph  -> 
(/)  e.  B )
Assertion
Ref Expression
cnfcom2lem  |-  ( 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 cnfcom2lem
StepHypRef Expression
1 cnfcom2.1 . . . . . 6  |-  ( ph  -> 
(/)  e.  B )
2 n0i 3460 . . . . . 6  |-  ( (/)  e.  B  ->  -.  B  =  (/) )
31, 2syl 15 . . . . 5  |-  ( ph  ->  -.  B  =  (/) )
4 cnfcom.f . . . . . . . . . . . . . 14  |-  F  =  ( `' ( om CNF 
A ) `  B
)
5 cnfcom.s . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( om CNF  A
)
6 omelon 7347 . . . . . . . . . . . . . . . . . 18  |-  om  e.  On
76a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  om  e.  On )
8 cnfcom.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
95, 7, 8cantnff1o 7398 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
) )
10 f1ocnv 5485 . . . . . . . . . . . . . . . 16  |-  ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  ->  `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S )
11 f1of 5472 . . . . . . . . . . . . . . . 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 cnfcom.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  ( om 
^o  A ) )
14 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( `' ( om CNF  A
) : ( om 
^o  A ) --> S  /\  B  e.  ( om  ^o  A ) )  ->  ( `' ( om CNF  A ) `  B )  e.  S
)
1512, 13, 14syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( om CNF 
A ) `  B
)  e.  S )
164, 15syl5eqel 2367 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  S )
175, 7, 8cantnfs 7367 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  e.  S  <->  ( F : A --> om  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
1816, 17mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( F : A --> om  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
1918simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  F : A --> om )
2019adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F : A
--> om )
2120feqmptd 5575 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
22 dif0 3524 . . . . . . . . . . . 12  |-  ( A 
\  (/) )  =  A
2322eleq2i 2347 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  (/) )  <->  x  e.  A
)
24 df1o2 6491 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
2524difeq2i 3291 . . . . . . . . . . . . . . 15  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2625imaeq2i 5010 . . . . . . . . . . . . . 14  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
27 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  =  (/) )
28 cnvimass 5033 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
29 fdm 5393 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : A --> om  ->  dom 
F  =  A )
3019, 29syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  F  =  A )
3128, 30syl5sseq 3226 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  A
)
32 ssexg 4160 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( `' F "
( _V  \  1o ) )  C_  A  /\  A  e.  On )  ->  ( `' F " ( _V  \  1o ) )  e.  _V )
3331, 8, 32syl2anc 642 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
34 cnfcom.g . . . . . . . . . . . . . . . . . . . . 21  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
355, 7, 8, 34, 16cantnfcl 7368 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
3635simpld 445 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
3734oien 7253 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3833, 36, 37syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3938adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  ~~  ( `' F "
( _V  \  1o ) ) )
4027, 39eqbrtrrd 4045 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  dom  G  =  (/) )  ->  (/)  ~~  ( `' F " ( _V 
\  1o ) ) )
41 ensym 6910 . . . . . . . . . . . . . . . 16  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
4240, 41syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
43 en0 6924 . . . . . . . . . . . . . . 15  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
4442, 43sylib 188 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) )  =  (/) )
4526, 44syl5eqr 2329 . . . . . . . . . . . . 13  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  =  (/) )
46 ss0b 3484 . . . . . . . . . . . . 13  |-  ( ( `' F " ( _V 
\  { (/) } ) )  C_  (/)  <->  ( `' F " ( _V  \  { (/) } ) )  =  (/) )
4745, 46sylibr 203 . . . . . . . . . . . 12  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  C_  (/) )
4820, 47suppssr 5659 . . . . . . . . . . 11  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  ( A  \  (/) ) )  ->  ( F `  x )  =  (/) )
4923, 48sylan2br 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  (/) )
5049mpteq2dva 4106 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  (/) ) )
5121, 50eqtrd 2315 . . . . . . . 8  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  (/) ) )
52 fconstmpt 4732 . . . . . . . 8  |-  ( A  X.  { (/) } )  =  ( x  e.  A  |->  (/) )
5351, 52syl6eqr 2333 . . . . . . 7  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( A  X.  { (/)
} ) )
5453fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( A  X.  { (/)
} ) ) )
554fveq2i 5528 . . . . . . . 8  |-  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( `' ( om CNF 
A ) `  B
) )
56 f1ocnvfv2 5793 . . . . . . . . 9  |-  ( ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  /\  B  e.  ( om  ^o  A ) )  ->  ( ( om CNF  A ) `  ( `' ( om CNF  A
) `  B )
)  =  B )
579, 13, 56syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( om CNF  A
) `  ( `' ( om CNF  A ) `  B ) )  =  B )
5855, 57syl5eq 2327 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  F )  =  B )
5958adantr 451 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  B )
60 peano1 4675 . . . . . . . . 9  |-  (/)  e.  om
6160a1i 10 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  om )
625, 7, 8, 61cantnf0 7376 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  ( A  X.  { (/) } ) )  =  (/) )
6362adantr 451 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  ( A  X.  { (/) } ) )  =  (/) )
6454, 59, 633eqtr3d 2323 . . . . 5  |-  ( (
ph  /\  dom  G  =  (/) )  ->  B  =  (/) )
653, 64mtand 640 . . . 4  |-  ( ph  ->  -.  dom  G  =  (/) )
6635simprd 449 . . . . 5  |-  ( ph  ->  dom  G  e.  om )
67 nnlim 4669 . . . . 5  |-  ( dom 
G  e.  om  ->  -. 
Lim  dom  G )
6866, 67syl 15 . . . 4  |-  ( ph  ->  -.  Lim  dom  G
)
69 ioran 476 . . . 4  |-  ( -.  ( dom  G  =  (/)  \/  Lim  dom  G
)  <->  ( -.  dom  G  =  (/)  /\  -.  Lim  dom 
G ) )
7065, 68, 69sylanbrc 645 . . 3  |-  ( ph  ->  -.  ( dom  G  =  (/)  \/  Lim  dom  G ) )
7134oicl 7244 . . . 4  |-  Ord  dom  G
72 unizlim 4509 . . . 4  |-  ( Ord 
dom  G  ->  ( dom 
G  =  U. dom  G  <-> 
( dom  G  =  (/) 
\/  Lim  dom  G ) ) )
7371, 72ax-mp 8 . . 3  |-  ( dom 
G  =  U. dom  G  <-> 
( dom  G  =  (/) 
\/  Lim  dom  G ) )
7470, 73sylnibr 296 . 2  |-  ( ph  ->  -.  dom  G  = 
U. dom  G )
75 orduniorsuc 4621 . . . 4  |-  ( Ord 
dom  G  ->  ( dom 
G  =  U. dom  G  \/  dom  G  =  suc  U. dom  G
) )
7671, 75mp1i 11 . . 3  |-  ( ph  ->  ( dom  G  = 
U. dom  G  \/  dom  G  =  suc  U. dom  G ) )
7776ord 366 . 2  |-  ( ph  ->  ( -.  dom  G  =  U. dom  G  ->  dom  G  =  suc  U. dom  G ) )
7874, 77mpd 14 1  |-  ( ph  ->  dom  G  =  suc  U.
dom  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    _E cep 4303    We wwe 4351   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459   1oc1o 6472    +o coa 6476    .o comu 6477    ^o coe 6478    ~~ cen 6860   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cnfcom2  7405  cnfcom3lem  7406  cnfcom3  7407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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