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Theorem cnfcom2lem 7420
Description: Lemma for cnfcom2 7421. (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 3473 . . . . . 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 7363 . . . . . . . . . . . . . . . . . 18  |-  om  e.  On
76a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  om  e.  On )
8 cnfcom.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
95, 7, 8cantnff1o 7414 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
) )
10 f1ocnv 5501 . . . . . . . . . . . . . . . 16  |-  ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  ->  `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S )
11 f1of 5488 . . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . 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 2380 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  S )
175, 7, 8cantnfs 7383 . . . . . . . . . . . . 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 5591 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
22 dif0 3537 . . . . . . . . . . . 12  |-  ( A 
\  (/) )  =  A
2322eleq2i 2360 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  (/) )  <->  x  e.  A
)
24 df1o2 6507 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
2524difeq2i 3304 . . . . . . . . . . . . . . 15  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2625imaeq2i 5026 . . . . . . . . . . . . . 14  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
27 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  =  (/) )
28 cnvimass 5049 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
29 fdm 5409 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : A --> om  ->  dom 
F  =  A )
3019, 29syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  F  =  A )
3128, 30syl5sseq 3239 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  A
)
32 ssexg 4176 . . . . . . . . . . . . . . . . . . . 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 7384 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
3635simpld 445 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
3734oien 7269 . . . . . . . . . . . . . . . . . . 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 4061 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  dom  G  =  (/) )  ->  (/)  ~~  ( `' F " ( _V 
\  1o ) ) )
41 ensym 6926 . . . . . . . . . . . . . . . 16  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
4240, 41syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
43 en0 6940 . . . . . . . . . . . . . . 15  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
4442, 43sylib 188 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) )  =  (/) )
4526, 44syl5eqr 2342 . . . . . . . . . . . . 13  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  =  (/) )
46 ss0b 3497 . . . . . . . . . . . . 13  |-  ( ( `' F " ( _V 
\  { (/) } ) )  C_  (/)  <->  ( `' F " ( _V  \  { (/) } ) )  =  (/) )
4745, 46sylibr 203 . . . . . . . . . . . 12  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  C_  (/) )
4820, 47suppssr 5675 . . . . . . . . . . 11  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  ( A  \  (/) ) )  ->  ( F `  x )  =  (/) )
4923, 48sylan2br 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  (/) )
5049mpteq2dva 4122 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  (/) ) )
5121, 50eqtrd 2328 . . . . . . . 8  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  (/) ) )
52 fconstmpt 4748 . . . . . . . 8  |-  ( A  X.  { (/) } )  =  ( x  e.  A  |->  (/) )
5351, 52syl6eqr 2346 . . . . . . 7  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( A  X.  { (/)
} ) )
5453fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( A  X.  { (/)
} ) ) )
554fveq2i 5544 . . . . . . . 8  |-  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( `' ( om CNF 
A ) `  B
) )
56 f1ocnvfv2 5809 . . . . . . . . 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 2340 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  F )  =  B )
5958adantr 451 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  B )
60 peano1 4691 . . . . . . . . 9  |-  (/)  e.  om
6160a1i 10 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  om )
625, 7, 8, 61cantnf0 7392 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  ( A  X.  { (/) } ) )  =  (/) )
6362adantr 451 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  ( A  X.  { (/) } ) )  =  (/) )
6454, 59, 633eqtr3d 2336 . . . . 5  |-  ( (
ph  /\  dom  G  =  (/) )  ->  B  =  (/) )
653, 64mtand 640 . . . 4  |-  ( ph  ->  -.  dom  G  =  (/) )
6635simprd 449 . . . . 5  |-  ( ph  ->  dom  G  e.  om )
67 nnlim 4685 . . . . 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 7260 . . . 4  |-  Ord  dom  G
72 unizlim 4525 . . . 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 4637 . . . 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093    _E cep 4319    We wwe 4367   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410   omcom 4672    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876  seq𝜔cseqom 6475   1oc1o 6488    +o coa 6492    .o comu 6493    ^o coe 6494    ~~ cen 6876   Fincfn 6879  OrdIsocoi 7240   CNF ccnf 7378
This theorem is referenced by:  cnfcom2  7421  cnfcom3lem  7422  cnfcom3  7423
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  ax-inf2 7358
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-rmo 2564  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-int 3879  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379
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