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Theorem cantnflem1bOLD 8021
Description: Lemma for cantnfOLD 8027. (Contributed by Mario Carneiro, 4-Jun-2015.) Obsolete version of cantnflem1a 7997 as of 2-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
oemapvalOLD.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapvalOLD.3  |-  ( ph  ->  F  e.  S )
oemapvalOLD.4  |-  ( ph  ->  G  e.  S )
oemapvalOLD.5  |-  ( ph  ->  F T G )
oemapvalOLD.6  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1OLD.o  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
Assertion
Ref Expression
cantnflem1bOLD  |-  ( (
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 cantnflem1bOLD
StepHypRef Expression
1 simprr 756 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  C_  u
)
2 cantnflem1OLD.o . . . . . . . 8  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
32oicl 7847 . . . . . . 7  |-  Ord  dom  O
4 cantnfsOLD.3 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  On )
5 cnvimass 5290 . . . . . . . . . . . . 13  |-  ( `' G " ( _V 
\  1o ) ) 
C_  dom  G
6 oemapvalOLD.4 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  S )
7 cantnfsOLD.1 . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( A CNF  B
)
8 cantnfsOLD.2 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
97, 8, 4cantnfsOLD 8008 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
106, 9mpbid 210 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
1110simpld 459 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : B --> A )
12 fdm 5664 . . . . . . . . . . . . . 14  |-  ( G : B --> A  ->  dom  G  =  B )
1311, 12syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  G  =  B )
145, 13syl5sseq 3505 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  B
)
154, 14ssexd 4540 . . . . . . . . . . 11  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  _V )
167, 8, 4, 2, 6cantnfclOLD 8009 . . . . . . . . . . . 12  |-  ( ph  ->  (  _E  We  ( `' G " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
1716simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  _E  We  ( `' G " ( _V 
\  1o ) ) )
182oiiso 7855 . . . . . . . . . . 11  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' G " ( _V 
\  1o ) ) )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ( `' G " ( _V 
\  1o ) ) ) )
1915, 17, 18syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  O  Isom  _E  ,  _E  ( dom  O ,  ( `' G " ( _V 
\  1o ) ) ) )
20 isof1o 6118 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ( `' G " ( _V 
\  1o ) ) )  ->  O : dom  O -1-1-onto-> ( `' G "
( _V  \  1o ) ) )
2119, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) ) )
22 f1ocnv 5754 . . . . . . . . 9  |-  ( O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) )  ->  `' O :
( `' G "
( _V  \  1o ) ) -1-1-onto-> dom  O )
23 f1of 5742 . . . . . . . . 9  |-  ( `' O : ( `' G " ( _V 
\  1o ) ) -1-1-onto-> dom 
O  ->  `' O : ( `' G " ( _V  \  1o ) ) --> dom  O
)
2421, 22, 233syl 20 . . . . . . . 8  |-  ( ph  ->  `' O : ( `' G " ( _V 
\  1o ) ) --> dom  O )
25 oemapvalOLD.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
) ) ) }
26 oemapvalOLD.3 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
27 oemapvalOLD.5 . . . . . . . . 9  |-  ( ph  ->  F T G )
28 oemapvalOLD.6 . . . . . . . . 9  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
297, 8, 4, 25, 26, 6, 27, 28cantnflem1aOLD 8020 . . . . . . . 8  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
3024, 29ffvelrnd 5946 . . . . . . 7  |-  ( ph  ->  ( `' O `  X )  e.  dom  O )
31 ordelon 4844 . . . . . . 7  |-  ( ( Ord  dom  O  /\  ( `' O `  X )  e.  dom  O )  ->  ( `' O `  X )  e.  On )
323, 30, 31sylancr 663 . . . . . 6  |-  ( ph  ->  ( `' O `  X )  e.  On )
3332adantr 465 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  On )
343a1i 11 . . . . . . . 8  |-  ( ph  ->  Ord  dom  O )
35 ordelon 4844 . . . . . . . 8  |-  ( ( Ord  dom  O  /\  suc  u  e.  dom  O
)  ->  suc  u  e.  On )
3634, 35sylan 471 . . . . . . 7  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  suc  u  e.  On )
37 sucelon 6531 . . . . . . 7  |-  ( u  e.  On  <->  suc  u  e.  On )
3836, 37sylibr 212 . . . . . 6  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  u  e.  On )
3938adantrr 716 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  On )
40 ontri1 4854 . . . . 5  |-  ( ( ( `' O `  X )  e.  On  /\  u  e.  On )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
4133, 39, 40syl2anc 661 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
421, 41mpbid 210 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  u  e.  ( `' O `  X ) )
4319adantr 465 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( `' G "
( _V  \  1o ) ) ) )
44 ordtr 4834 . . . . . . . 8  |-  ( Ord 
dom  O  ->  Tr  dom  O )
453, 44mp1i 12 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  Tr  dom  O )
46 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  suc  u  e.  dom  O )
47 trsuc 4904 . . . . . . 7  |-  ( ( Tr  dom  O  /\  suc  u  e.  dom  O
)  ->  u  e.  dom  O )
4845, 46, 47syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  dom  O )
4930adantr 465 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  dom  O )
50 isorel 6119 . . . . . 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 )
) ) )
5143, 48, 49, 50syl12anc 1217 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
52 fvex 5802 . . . . . 6  |-  ( `' O `  X )  e.  _V
5352epelc 4735 . . . . 5  |-  ( u  _E  ( `' O `  X )  <->  u  e.  ( `' O `  X ) )
54 fvex 5802 . . . . . 6  |-  ( O `
 ( `' O `  X ) )  e. 
_V
5554epelc 4735 . . . . 5  |-  ( ( O `  u )  _E  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) )
5651, 53, 553bitr3g 287 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) ) )
57 f1ocnvfv2 6086 . . . . . . 7  |-  ( ( O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) )  /\  X  e.  ( `' G " ( _V 
\  1o ) ) )  ->  ( O `  ( `' O `  X ) )  =  X )
5821, 29, 57syl2anc 661 . . . . . 6  |-  ( ph  ->  ( O `  ( `' O `  X ) )  =  X )
5958adantr 465 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6059eleq2d 2521 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( O `
 u )  e.  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  X
) )
6156, 60bitrd 253 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  X
) )
6242, 61mtbid 300 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  ( O `  u )  e.  X
)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 7996 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
6463simp1d 1000 . . . . 5  |-  ( ph  ->  X  e.  B )
65 onelon 4845 . . . . 5  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
664, 64, 65syl2anc 661 . . . 4  |-  ( ph  ->  X  e.  On )
6766adantr 465 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  e.  On )
684adantr 465 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  B  e.  On )
6914adantr 465 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' G " ( _V  \  1o ) )  C_  B
)
702oif 7848 . . . . . . 7  |-  O : dom  O --> ( `' G " ( _V  \  1o ) )
7170ffvelrni 5944 . . . . . 6  |-  ( u  e.  dom  O  -> 
( O `  u
)  e.  ( `' G " ( _V 
\  1o ) ) )
7248, 71syl 16 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  ( `' G " ( _V 
\  1o ) ) )
7369, 72sseldd 3458 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  B
)
74 onelon 4845 . . . 4  |-  ( ( B  e.  On  /\  ( O `  u )  e.  B )  -> 
( O `  u
)  e.  On )
7568, 73, 74syl2anc 661 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  On )
76 ontri1 4854 . . 3  |-  ( ( X  e.  On  /\  ( O `  u )  e.  On )  -> 
( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7767, 75, 76syl2anc 661 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7862, 77mpbird 232 1  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3071    \ cdif 3426    C_ wss 3429   U.cuni 4192   class class class wbr 4393   {copab 4450   Tr wtr 4486    _E cep 4731    We wwe 4779   Ord word 4819   Oncon0 4820   suc csuc 4822   `'ccnv 4940   dom cdm 4941   "cima 4944   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519    Isom wiso 5520  (class class class)co 6193   omcom 6579   1oc1o 7016   Fincfn 7413  OrdIsocoi 7827   CNF ccnf 7971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-seqom 7006  df-1o 7023  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-cnf 7972
This theorem is referenced by:  cantnflem1cOLD  8022
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