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Theorem cantnflem1bOLD 8119
Description: Lemma for cantnfOLD 8125. (Contributed by Mario Carneiro, 4-Jun-2015.) Obsolete version of cantnflem1a 8095 as of 2-Jul-2019. (New usage is discouraged.) (Proof modification 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 755 . . . 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 7946 . . . . . . 7  |-  Ord  dom  O
4 cantnfsOLD.3 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  On )
5 cnvimass 5345 . . . . . . . . . . . . 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 8106 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : B --> A )
12 fdm 5717 . . . . . . . . . . . . . 14  |-  ( G : B --> A  ->  dom  G  =  B )
1311, 12syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  G  =  B )
145, 13syl5sseq 3537 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  B
)
154, 14ssexd 4584 . . . . . . . . . . 11  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  _V )
167, 8, 4, 2, 6cantnfclOLD 8107 . . . . . . . . . . . 12  |-  ( ph  ->  (  _E  We  ( `' G " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
1716simpld 457 . . . . . . . . . . 11  |-  ( ph  ->  _E  We  ( `' G " ( _V 
\  1o ) ) )
182oiiso 7954 . . . . . . . . . . 11  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' G " ( _V 
\  1o ) ) )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ( `' G " ( _V 
\  1o ) ) ) )
1915, 17, 18syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  O  Isom  _E  ,  _E  ( dom  O ,  ( `' G " ( _V 
\  1o ) ) ) )
20 isof1o 6196 . . . . . . . . . 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 5810 . . . . . . . . 9  |-  ( O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) )  ->  `' O :
( `' G "
( _V  \  1o ) ) -1-1-onto-> dom  O )
23 f1of 5798 . . . . . . . . 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 8118 . . . . . . . 8  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
3024, 29ffvelrnd 6008 . . . . . . 7  |-  ( ph  ->  ( `' O `  X )  e.  dom  O )
31 ordelon 4891 . . . . . . 7  |-  ( ( Ord  dom  O  /\  ( `' O `  X )  e.  dom  O )  ->  ( `' O `  X )  e.  On )
323, 30, 31sylancr 661 . . . . . 6  |-  ( ph  ->  ( `' O `  X )  e.  On )
3332adantr 463 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  On )
343a1i 11 . . . . . . . 8  |-  ( ph  ->  Ord  dom  O )
35 ordelon 4891 . . . . . . . 8  |-  ( ( Ord  dom  O  /\  suc  u  e.  dom  O
)  ->  suc  u  e.  On )
3634, 35sylan 469 . . . . . . 7  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  suc  u  e.  On )
37 sucelon 6625 . . . . . . 7  |-  ( u  e.  On  <->  suc  u  e.  On )
3836, 37sylibr 212 . . . . . 6  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  u  e.  On )
3938adantrr 714 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  On )
40 ontri1 4901 . . . . 5  |-  ( ( ( `' O `  X )  e.  On  /\  u  e.  On )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
4133, 39, 40syl2anc 659 . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( `' G "
( _V  \  1o ) ) ) )
44 ordtr 4881 . . . . . . . 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 754 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  suc  u  e.  dom  O )
47 trsuc 4951 . . . . . . 7  |-  ( ( Tr  dom  O  /\  suc  u  e.  dom  O
)  ->  u  e.  dom  O )
4845, 46, 47syl2anc 659 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  dom  O )
4930adantr 463 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  dom  O )
50 isorel 6197 . . . . . 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 1224 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
52 fvex 5858 . . . . . 6  |-  ( `' O `  X )  e.  _V
5352epelc 4782 . . . . 5  |-  ( u  _E  ( `' O `  X )  <->  u  e.  ( `' O `  X ) )
54 fvex 5858 . . . . . 6  |-  ( O `
 ( `' O `  X ) )  e. 
_V
5554epelc 4782 . . . . 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 6158 . . . . . . 7  |-  ( ( O : dom  O -1-1-onto-> ( `' G " ( _V 
\  1o ) )  /\  X  e.  ( `' G " ( _V 
\  1o ) ) )  ->  ( O `  ( `' O `  X ) )  =  X )
5821, 29, 57syl2anc 659 . . . . . 6  |-  ( ph  ->  ( O `  ( `' O `  X ) )  =  X )
5958adantr 463 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6059eleq2d 2524 . . . 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 298 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  ( O `  u )  e.  X
)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 8094 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
6463simp1d 1006 . . . . 5  |-  ( ph  ->  X  e.  B )
65 onelon 4892 . . . . 5  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
664, 64, 65syl2anc 659 . . . 4  |-  ( ph  ->  X  e.  On )
6766adantr 463 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  e.  On )
684adantr 463 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  B  e.  On )
6914adantr 463 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' G " ( _V  \  1o ) )  C_  B
)
702oif 7947 . . . . . . 7  |-  O : dom  O --> ( `' G " ( _V  \  1o ) )
7170ffvelrni 6006 . . . . . 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 3490 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  B
)
74 onelon 4892 . . . 4  |-  ( ( B  e.  On  /\  ( O `  u )  e.  B )  -> 
( O `  u
)  e.  On )
7568, 73, 74syl2anc 659 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  On )
76 ontri1 4901 . . 3  |-  ( ( X  e.  On  /\  ( O `  u )  e.  On )  -> 
( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7767, 75, 76syl2anc 659 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   U.cuni 4235   class class class wbr 4439   {copab 4496   Tr wtr 4532    _E cep 4778    We wwe 4826   Ord word 4866   Oncon0 4867   suc csuc 4869   `'ccnv 4987   dom cdm 4988   "cima 4991   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571  (class class class)co 6270   omcom 6673   1oc1o 7115   Fincfn 7509  OrdIsocoi 7926   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-cnf 8070
This theorem is referenced by:  cantnflem1cOLD  8120
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