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Theorem cantnflem1b 8209
Description: Lemma for cantnf 8216. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( 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.f  |-  ( ph  ->  F  e.  S )
oemapval.g  |-  ( ph  ->  G  e.  S )
oemapvali.r  |-  ( ph  ->  F T G )
oemapvali.x  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1.o  |-  O  = OrdIso
(  _E  ,  ( G supp  (/) ) )
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 774 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  C_  u
)
2 cantnflem1.o . . . . . . . 8  |-  O  = OrdIso
(  _E  ,  ( G supp  (/) ) )
32oicl 8062 . . . . . . 7  |-  Ord  dom  O
4 cantnfs.b . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  On )
5 suppssdm 6946 . . . . . . . . . . . . 13  |-  ( G supp  (/) )  C_  dom  G
6 oemapval.g . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  S )
7 cantnfs.s . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( A CNF  B
)
8 cantnfs.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
97, 8, 4cantnfs 8189 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
106, 9mpbid 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1110simpld 466 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : B --> A )
12 fdm 5745 . . . . . . . . . . . . . 14  |-  ( G : B --> A  ->  dom  G  =  B )
1311, 12syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  G  =  B )
145, 13syl5sseq 3466 . . . . . . . . . . . 12  |-  ( ph  ->  ( G supp  (/) )  C_  B )
154, 14ssexd 4543 . . . . . . . . . . 11  |-  ( ph  ->  ( G supp  (/) )  e. 
_V )
167, 8, 4, 2, 6cantnfcl 8190 . . . . . . . . . . . 12  |-  ( ph  ->  (  _E  We  ( G supp 
(/) )  /\  dom  O  e.  om ) )
1716simpld 466 . . . . . . . . . . 11  |-  ( ph  ->  _E  We  ( G supp  (/) ) )
182oiiso 8070 . . . . . . . . . . 11  |-  ( ( ( G supp  (/) )  e. 
_V  /\  _E  We  ( G supp  (/) ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( G supp  (/) ) ) )
1915, 17, 18syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  O  Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) ) )
20 isof1o 6234 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) )  ->  O : dom  O -1-1-onto-> ( G supp  (/) ) )
2119, 20syl 17 . . . . . . . . 9  |-  ( ph  ->  O : dom  O -1-1-onto-> ( G supp 
(/) ) )
22 f1ocnv 5840 . . . . . . . . 9  |-  ( O : dom  O -1-1-onto-> ( G supp  (/) )  ->  `' O : ( G supp  (/) ) -1-1-onto-> dom  O
)
23 f1of 5828 . . . . . . . . 9  |-  ( `' O : ( G supp  (/) ) -1-1-onto-> dom  O  ->  `' O : ( G supp  (/) ) --> dom 
O )
2421, 22, 233syl 18 . . . . . . . 8  |-  ( ph  ->  `' O : ( G supp  (/) ) --> dom  O )
25 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
) ) ) }
26 oemapval.f . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
27 oemapvali.r . . . . . . . . 9  |-  ( ph  ->  F T G )
28 oemapvali.x . . . . . . . . 9  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
297, 8, 4, 25, 26, 6, 27, 28cantnflem1a 8208 . . . . . . . 8  |-  ( ph  ->  X  e.  ( G supp  (/) ) )
3024, 29ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( `' O `  X )  e.  dom  O )
31 ordelon 5454 . . . . . . 7  |-  ( ( Ord  dom  O  /\  ( `' O `  X )  e.  dom  O )  ->  ( `' O `  X )  e.  On )
323, 30, 31sylancr 676 . . . . . 6  |-  ( ph  ->  ( `' O `  X )  e.  On )
3332adantr 472 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  On )
343a1i 11 . . . . . . . 8  |-  ( ph  ->  Ord  dom  O )
35 ordelon 5454 . . . . . . . 8  |-  ( ( Ord  dom  O  /\  suc  u  e.  dom  O
)  ->  suc  u  e.  On )
3634, 35sylan 479 . . . . . . 7  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  suc  u  e.  On )
37 sucelon 6663 . . . . . . 7  |-  ( u  e.  On  <->  suc  u  e.  On )
3836, 37sylibr 217 . . . . . 6  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  u  e.  On )
3938adantrr 731 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  On )
40 ontri1 5464 . . . . 5  |-  ( ( ( `' O `  X )  e.  On  /\  u  e.  On )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
4133, 39, 40syl2anc 673 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
421, 41mpbid 215 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  u  e.  ( `' O `  X ) )
4319adantr 472 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( G supp  (/) ) ) )
44 ordtr 5444 . . . . . . . 8  |-  ( Ord 
dom  O  ->  Tr  dom  O )
453, 44mp1i 13 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  Tr  dom  O )
46 simprl 772 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  suc  u  e.  dom  O )
47 trsuc 5514 . . . . . . 7  |-  ( ( Tr  dom  O  /\  suc  u  e.  dom  O
)  ->  u  e.  dom  O )
4845, 46, 47syl2anc 673 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  dom  O )
4930adantr 472 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  dom  O )
50 isorel 6235 . . . . . 6  |-  ( ( O  Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) )  /\  ( u  e.  dom  O  /\  ( `' O `  X )  e.  dom  O ) )  ->  (
u  _E  ( `' O `  X )  <-> 
( O `  u
)  _E  ( O `
 ( `' O `  X ) ) ) )
5143, 48, 49, 50syl12anc 1290 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
52 fvex 5889 . . . . . 6  |-  ( `' O `  X )  e.  _V
5352epelc 4752 . . . . 5  |-  ( u  _E  ( `' O `  X )  <->  u  e.  ( `' O `  X ) )
54 fvex 5889 . . . . . 6  |-  ( O `
 ( `' O `  X ) )  e. 
_V
5554epelc 4752 . . . . 5  |-  ( ( O `  u )  _E  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) )
5651, 53, 553bitr3g 295 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) ) )
57 f1ocnvfv2 6194 . . . . . . 7  |-  ( ( O : dom  O -1-1-onto-> ( G supp 
(/) )  /\  X  e.  ( G supp  (/) ) )  ->  ( O `  ( `' O `  X ) )  =  X )
5821, 29, 57syl2anc 673 . . . . . 6  |-  ( ph  ->  ( O `  ( `' O `  X ) )  =  X )
5958adantr 472 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6059eleq2d 2534 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( O `
 u )  e.  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  X
) )
6156, 60bitrd 261 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  X
) )
6242, 61mtbid 307 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  ( O `  u )  e.  X
)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 8207 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
6463simp1d 1042 . . . . 5  |-  ( ph  ->  X  e.  B )
65 onelon 5455 . . . . 5  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
664, 64, 65syl2anc 673 . . . 4  |-  ( ph  ->  X  e.  On )
6766adantr 472 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  e.  On )
684adantr 472 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  B  e.  On )
6914adantr 472 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( G supp  (/) )  C_  B )
702oif 8063 . . . . . . 7  |-  O : dom  O --> ( G supp  (/) )
7170ffvelrni 6036 . . . . . 6  |-  ( u  e.  dom  O  -> 
( O `  u
)  e.  ( G supp  (/) ) )
7248, 71syl 17 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  ( G supp  (/) ) )
7369, 72sseldd 3419 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  B
)
74 onelon 5455 . . . 4  |-  ( ( B  e.  On  /\  ( O `  u )  e.  B )  -> 
( O `  u
)  e.  On )
7568, 73, 74syl2anc 673 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  On )
76 ontri1 5464 . . 3  |-  ( ( X  e.  On  /\  ( O `  u )  e.  On )  -> 
( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7767, 75, 76syl2anc 673 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7862, 77mpbird 240 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   U.cuni 4190   class class class wbr 4395   {copab 4453   Tr wtr 4490    _E cep 4748    We wwe 4797   `'ccnv 4838   dom cdm 4839   Ord word 5429   Oncon0 5430   suc csuc 5432   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   omcom 6711   supp csupp 6933   finSupp cfsupp 7901  OrdIsocoi 8042   CNF ccnf 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seqom 7183  df-1o 7200  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-cnf 8185
This theorem is referenced by:  cantnflem1c  8210
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