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Theorem cantnflem1b 7404
Description: Lemma for cantnf 7411. (Contributed by Mario Carneiro, 4-Jun-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapval.3  |-  ( ph  ->  F  e.  S )
oemapval.4  |-  ( ph  ->  G  e.  S )
oemapvali.5  |-  ( ph  ->  F T G )
oemapvali.6  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1.o  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
Assertion
Ref Expression
cantnflem1b  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
Distinct variable groups:    u, c, w, x, y, z, B    A, c, u, w, x, y, z    T, c, u    u, F, w, x, y, z    S, c, u, x, y, z    G, c, u, w, x, y, z    u, O, w, x, y, z    ph, u, x, y, z   
u, X, w, x, y, z    F, c    ph, c
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)    O( c)    X( c)

Proof of Theorem cantnflem1b
StepHypRef Expression
1 simprr 733 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  C_  u
)
2 cantnflem1.o . . . . . . . 8  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
32oicl 7260 . . . . . . 7  |-  Ord  dom  O
4 cnvimass 5049 . . . . . . . . . . . . 13  |-  ( `' G " ( _V 
\  1o ) ) 
C_  dom  G
5 oemapval.4 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  S )
6 cantnfs.1 . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( A CNF  B
)
7 cantnfs.2 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
8 cantnfs.3 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  e.  On )
96, 7, 8cantnfs 7383 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
105, 9mpbid 201 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
1110simpld 445 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : B --> A )
12 fdm 5409 . . . . . . . . . . . . . 14  |-  ( G : B --> A  ->  dom  G  =  B )
1311, 12syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  G  =  B )
144, 13syl5sseq 3239 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  B
)
15 ssexg 4176 . . . . . . . . . . . 12  |-  ( ( ( `' G "
( _V  \  1o ) )  C_  B  /\  B  e.  On )  ->  ( `' G " ( _V  \  1o ) )  e.  _V )
1614, 8, 15syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  _V )
176, 7, 8, 2, 5cantnfcl 7384 . . . . . . . . . . . 12  |-  ( ph  ->  (  _E  We  ( `' G " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
1817simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  _E  We  ( `' G " ( _V 
\  1o ) ) )
192oiiso 7268 . . . . . . . . . . 11  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' G " ( _V 
\  1o ) ) )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ( `' G " ( _V 
\  1o ) ) ) )
2016, 18, 19syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  O  Isom  _E  ,  _E  ( dom  O ,  ( `' G " ( _V 
\  1o ) ) ) )
21 isof1o 5838 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ( `' G " ( _V 
\  1o ) ) )  ->  O : dom  O -1-1-onto-> ( `' G "
( _V  \  1o ) ) )
2220, 21syl 15 . . . . . . . . 9  |-  ( ph  ->  O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) ) )
23 f1ocnv 5501 . . . . . . . . 9  |-  ( O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) )  ->  `' O :
( `' G "
( _V  \  1o ) ) -1-1-onto-> dom  O )
24 f1of 5488 . . . . . . . . 9  |-  ( `' O : ( `' G " ( _V 
\  1o ) ) -1-1-onto-> dom 
O  ->  `' O : ( `' G " ( _V  \  1o ) ) --> dom  O
)
2522, 23, 243syl 18 . . . . . . . 8  |-  ( ph  ->  `' O : ( `' G " ( _V 
\  1o ) ) --> dom  O )
26 oemapval.t . . . . . . . . 9  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
27 oemapval.3 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
28 oemapvali.5 . . . . . . . . 9  |-  ( ph  ->  F T G )
29 oemapvali.6 . . . . . . . . 9  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
306, 7, 8, 26, 27, 5, 28, 29cantnflem1a 7403 . . . . . . . 8  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
31 ffvelrn 5679 . . . . . . . 8  |-  ( ( `' O : ( `' G " ( _V 
\  1o ) ) --> dom  O  /\  X  e.  ( `' G "
( _V  \  1o ) ) )  -> 
( `' O `  X )  e.  dom  O )
3225, 30, 31syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( `' O `  X )  e.  dom  O )
33 ordelon 4432 . . . . . . 7  |-  ( ( Ord  dom  O  /\  ( `' O `  X )  e.  dom  O )  ->  ( `' O `  X )  e.  On )
343, 32, 33sylancr 644 . . . . . 6  |-  ( ph  ->  ( `' O `  X )  e.  On )
3534adantr 451 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  On )
363a1i 10 . . . . . . . 8  |-  ( ph  ->  Ord  dom  O )
37 ordelon 4432 . . . . . . . 8  |-  ( ( Ord  dom  O  /\  suc  u  e.  dom  O
)  ->  suc  u  e.  On )
3836, 37sylan 457 . . . . . . 7  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  suc  u  e.  On )
39 sucelon 4624 . . . . . . 7  |-  ( u  e.  On  <->  suc  u  e.  On )
4038, 39sylibr 203 . . . . . 6  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  u  e.  On )
4140adantrr 697 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  On )
42 ontri1 4442 . . . . 5  |-  ( ( ( `' O `  X )  e.  On  /\  u  e.  On )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
4335, 41, 42syl2anc 642 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
441, 43mpbid 201 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  u  e.  ( `' O `  X ) )
4520adantr 451 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( `' G "
( _V  \  1o ) ) ) )
46 ordtr 4422 . . . . . . . 8  |-  ( Ord 
dom  O  ->  Tr  dom  O )
473, 46mp1i 11 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  Tr  dom  O )
48 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  suc  u  e.  dom  O )
49 trsuc 4492 . . . . . . 7  |-  ( ( Tr  dom  O  /\  suc  u  e.  dom  O
)  ->  u  e.  dom  O )
5047, 48, 49syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  dom  O )
5132adantr 451 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  dom  O )
52 isorel 5839 . . . . . 6  |-  ( ( O  Isom  _E  ,  _E  ( dom  O ,  ( `' G " ( _V 
\  1o ) ) )  /\  ( u  e.  dom  O  /\  ( `' O `  X )  e.  dom  O ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
5345, 50, 51, 52syl12anc 1180 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
54 fvex 5555 . . . . . 6  |-  ( `' O `  X )  e.  _V
5554epelc 4323 . . . . 5  |-  ( u  _E  ( `' O `  X )  <->  u  e.  ( `' O `  X ) )
56 fvex 5555 . . . . . 6  |-  ( O `
 ( `' O `  X ) )  e. 
_V
5756epelc 4323 . . . . 5  |-  ( ( O `  u )  _E  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) )
5853, 55, 573bitr3g 278 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) ) )
59 f1ocnvfv2 5809 . . . . . . 7  |-  ( ( O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) )  /\  X  e.  ( `' G " ( _V 
\  1o ) ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6022, 30, 59syl2anc 642 . . . . . 6  |-  ( ph  ->  ( O `  ( `' O `  X ) )  =  X )
6160adantr 451 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6261eleq2d 2363 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( O `
 u )  e.  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  X
) )
6358, 62bitrd 244 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  X
) )
6444, 63mtbid 291 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  ( O `  u )  e.  X
)
656, 7, 8, 26, 27, 5, 28, 29oemapvali 7402 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
6665simp1d 967 . . . . 5  |-  ( ph  ->  X  e.  B )
67 onelon 4433 . . . . 5  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
688, 66, 67syl2anc 642 . . . 4  |-  ( ph  ->  X  e.  On )
6968adantr 451 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  e.  On )
708adantr 451 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  B  e.  On )
7114adantr 451 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' G " ( _V  \  1o ) )  C_  B
)
722oif 7261 . . . . . . 7  |-  O : dom  O --> ( `' G " ( _V  \  1o ) )
7372ffvelrni 5680 . . . . . 6  |-  ( u  e.  dom  O  -> 
( O `  u
)  e.  ( `' G " ( _V 
\  1o ) ) )
7450, 73syl 15 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  ( `' G " ( _V 
\  1o ) ) )
7571, 74sseldd 3194 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  B
)
76 onelon 4433 . . . 4  |-  ( ( B  e.  On  /\  ( O `  u )  e.  B )  -> 
( O `  u
)  e.  On )
7770, 75, 76syl2anc 642 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  On )
78 ontri1 4442 . . 3  |-  ( ( X  e.  On  /\  ( O `  u )  e.  On )  -> 
( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7969, 77, 78syl2anc 642 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
8064, 79mpbird 223 1  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   U.cuni 3843   class class class wbr 4039   {copab 4092   Tr wtr 4129    _E cep 4319    We wwe 4367   Ord word 4407   Oncon0 4408   suc csuc 4410   omcom 4672   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   1oc1o 6488   Fincfn 6879  OrdIsocoi 7240   CNF ccnf 7378
This theorem is referenced by:  cantnflem1c  7405
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
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-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-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  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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