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Theorem cnfcom2lem 7650
Description: Lemma for cnfcom2 7651. (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 3625 . . . . . 6  |-  ( (/)  e.  B  ->  -.  B  =  (/) )
31, 2syl 16 . . . . 5  |-  ( ph  ->  -.  B  =  (/) )
4 cnfcom.f . . . . . . . . . . . . . 14  |-  F  =  ( `' ( om CNF 
A ) `  B
)
5 cnfcom.s . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( om CNF  A
)
6 omelon 7593 . . . . . . . . . . . . . . . . . 18  |-  om  e.  On
76a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  om  e.  On )
8 cnfcom.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
95, 7, 8cantnff1o 7644 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
) )
10 f1ocnv 5679 . . . . . . . . . . . . . . . 16  |-  ( ( om CNF  A ) : S -1-1-onto-> ( om  ^o  A
)  ->  `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S )
11 f1of 5666 . . . . . . . . . . . . . . . 16  |-  ( `' ( om CNF  A ) : ( om  ^o  A ) -1-1-onto-> S  ->  `' ( om CNF  A ) : ( om  ^o  A ) --> S )
129, 10, 113syl 19 . . . . . . . . . . . . . . 15  |-  ( ph  ->  `' ( om CNF  A
) : ( om 
^o  A ) --> S )
13 cnfcom.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  ( om 
^o  A ) )
1412, 13ffvelrnd 5863 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( om CNF 
A ) `  B
)  e.  S )
154, 14syl5eqel 2519 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  S )
165, 7, 8cantnfs 7613 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  e.  S  <->  ( F : A --> om  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
1715, 16mpbid 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( F : A --> om  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
1817simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  F : A --> om )
1918adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F : A
--> om )
2019feqmptd 5771 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21 dif0 3690 . . . . . . . . . . . 12  |-  ( A 
\  (/) )  =  A
2221eleq2i 2499 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  (/) )  <->  x  e.  A
)
23 df1o2 6728 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
2423difeq2i 3454 . . . . . . . . . . . . . . 15  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2524imaeq2i 5193 . . . . . . . . . . . . . 14  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
26 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  =  (/) )
27 cnvimass 5216 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
28 fdm 5587 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : A --> om  ->  dom 
F  =  A )
2918, 28syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  F  =  A )
3027, 29syl5sseq 3388 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  A
)
318, 30ssexd 4342 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
32 cnfcom.g . . . . . . . . . . . . . . . . . . . . 21  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
335, 7, 8, 32, 15cantnfcl 7614 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
3433simpld 446 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
3532oien 7499 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3631, 34, 35syl2anc 643 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
3736adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  dom  G  =  (/) )  ->  dom  G  ~~  ( `' F "
( _V  \  1o ) ) )
3826, 37eqbrtrrd 4226 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  dom  G  =  (/) )  ->  (/)  ~~  ( `' F " ( _V 
\  1o ) ) )
3938ensymd 7150 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) ) 
~~  (/) )
40 en0 7162 . . . . . . . . . . . . . . 15  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
4139, 40sylib 189 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  1o ) )  =  (/) )
4225, 41syl5eqr 2481 . . . . . . . . . . . . 13  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  =  (/) )
43 ss0b 3649 . . . . . . . . . . . . 13  |-  ( ( `' F " ( _V 
\  { (/) } ) )  C_  (/)  <->  ( `' F " ( _V  \  { (/) } ) )  =  (/) )
4442, 43sylibr 204 . . . . . . . . . . . 12  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( `' F " ( _V 
\  { (/) } ) )  C_  (/) )
4519, 44suppssr 5856 . . . . . . . . . . 11  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  ( A  \  (/) ) )  ->  ( F `  x )  =  (/) )
4622, 45sylan2br 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  dom  G  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  (/) )
4746mpteq2dva 4287 . . . . . . . . 9  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  (/) ) )
4820, 47eqtrd 2467 . . . . . . . 8  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( x  e.  A  |->  (/) ) )
49 fconstmpt 4913 . . . . . . . 8  |-  ( A  X.  { (/) } )  =  ( x  e.  A  |->  (/) )
5048, 49syl6eqr 2485 . . . . . . 7  |-  ( (
ph  /\  dom  G  =  (/) )  ->  F  =  ( A  X.  { (/)
} ) )
5150fveq2d 5724 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( A  X.  { (/)
} ) ) )
524fveq2i 5723 . . . . . . . 8  |-  ( ( om CNF  A ) `  F )  =  ( ( om CNF  A ) `  ( `' ( om CNF 
A ) `  B
) )
53 f1ocnvfv2 6007 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ph  ->  ( ( om CNF  A
) `  ( `' ( om CNF  A ) `  B ) )  =  B )
5552, 54syl5eq 2479 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  F )  =  B )
5655adantr 452 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  F )  =  B )
57 peano1 4856 . . . . . . . . 9  |-  (/)  e.  om
5857a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  om )
595, 7, 8, 58cantnf0 7622 . . . . . . 7  |-  ( ph  ->  ( ( om CNF  A
) `  ( A  X.  { (/) } ) )  =  (/) )
6059adantr 452 . . . . . 6  |-  ( (
ph  /\  dom  G  =  (/) )  ->  ( ( om CNF  A ) `  ( A  X.  { (/) } ) )  =  (/) )
6151, 56, 603eqtr3d 2475 . . . . 5  |-  ( (
ph  /\  dom  G  =  (/) )  ->  B  =  (/) )
623, 61mtand 641 . . . 4  |-  ( ph  ->  -.  dom  G  =  (/) )
6333simprd 450 . . . . 5  |-  ( ph  ->  dom  G  e.  om )
64 nnlim 4850 . . . . 5  |-  ( dom 
G  e.  om  ->  -. 
Lim  dom  G )
6563, 64syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  G
)
66 ioran 477 . . . 4  |-  ( -.  ( dom  G  =  (/)  \/  Lim  dom  G
)  <->  ( -.  dom  G  =  (/)  /\  -.  Lim  dom 
G ) )
6762, 65, 66sylanbrc 646 . . 3  |-  ( ph  ->  -.  ( dom  G  =  (/)  \/  Lim  dom  G ) )
6832oicl 7490 . . . 4  |-  Ord  dom  G
69 unizlim 4690 . . . 4  |-  ( Ord 
dom  G  ->  ( dom 
G  =  U. dom  G  <-> 
( dom  G  =  (/) 
\/  Lim  dom  G ) ) )
7068, 69ax-mp 8 . . 3  |-  ( dom 
G  =  U. dom  G  <-> 
( dom  G  =  (/) 
\/  Lim  dom  G ) )
7167, 70sylnibr 297 . 2  |-  ( ph  ->  -.  dom  G  = 
U. dom  G )
72 orduniorsuc 4802 . . . 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 367 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   U.cuni 4007   class class class wbr 4204    e. cmpt 4258    _E cep 4484    We wwe 4532   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   omcom 4837    X. cxp 4868   `'ccnv 4869   dom cdm 4870   "cima 4873   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075  seq𝜔cseqom 6696   1oc1o 6709    +o coa 6713    .o comu 6714    ^o coe 6715    ~~ cen 7098   Fincfn 7101  OrdIsocoi 7470   CNF ccnf 7608
This theorem is referenced by:  cnfcom2  7651  cnfcom3lem  7652  cnfcom3  7653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-oexp 6722  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-cnf 7609
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